(and yes, if you're wondering, I have played bw/sc2/LoL)
Reading through this thread (and seeing many in the past), the two major points of contention seem to be 1) Popularity, and 2) Skill.
LoL is currently the 2nd most popular game in Korean PC Bangs with almost 16% (http://www.gametrics.com/); Broodwar is 6th with 3.41%, and SC2 isn't even in the top 10. In terms of worldwide popularity, LoL currently has separate servers for: NA, EUW (EU West), EUNE (EU Nordic/East), TW (Taiwan), China (multiple servers), Korea, and SEA (Garena?). Between NA/EUW/EUNE alone, there are about 32 million registered accounts. (http://www.pcworld.com/article/257292/european_league_of_legends_game_players_have_their_account_data_compromised.html)
Clearly, LoL is an extremely popular game, but why? Basically a combination of factors that includes:
1) The F2P/Microtransaction model
Players can start free (and stay free to play) and if they enjoy the game and continue playing, can spend real money in the game store.
LoL's F2P model here differs slightly with DOTA2. In DOTA2, you get every single champion for free and can only spend money buying cosmetic purchases (skins, different looking items, etc) that have no impact on the game. LoL on the other hand has a limited champion pool (10 champions free, rotating on a weekly basis) free. You can buy champions with IP/RP (Influence points - in game points that you get for playing games, and RP - riot points, bought with real money). Having a large champion pool can be an asset, but your playing ability matters far more. A bw/sc2 analogy would be the player who can execute many strategies well and can get to whatever rank versus the player who executes a single strategy extremely well and reaches the same rank.
Additionally, LoL has "Runes" which give you a small advantage in game - while runes are only purchasable with IP, "Rune pages" (basically a specific page of combined runes - you start off with 2) can be bought with RP. At the highest levels of play, you are somewhat constrained if you only have these default 2 rune pages available, as playing different champions optimally requires different rune setups.
2) The noob-friendly aspect of LoL
LoL has been designed from Day 1 to be appealing to as many people as possible. The graphics, computer requirements, and learning curve all reflect this. Anyone can jump into LoL. This is not a bad thing. Having a casual-friendly game attracts and (more importantly) holds onto players. If someone were to start playing Broodwar now with the intention to be even D+ on iccup, think about how hard even this modest accomplishment would be. They would have to:
1. Play through the single player campaign to familiarize themselves with the game UI and units.
2. Play 1v1 against a computer until they can beat it.
3. Read, watch and learn about every basic strategy and matchup involving their chosen race.
4. Practice executing these strategies versus real opponents on ICCUP (most likely losing 95%+ of the time).
Repeat steps 3+4.
How quickly do you think this hypothetical person would get fed up with the game and quit? Learning SC2 is a similar struggle. Having an excellent bw pedigree does not equate to instant success at SC2 - even if Flash plays SC2 full time, it may take him a few months before he reaches the absolute top of the game. Extrapolate that line of thought for the average person again: for someone who has never played Broodwar or SC2, imagine how hard it would be for them to get to Platinum in SC2 - they would end up being in Bronze for an entire season or two before getting the hang of the game (if they haven't quit).
LoL on the other hand, provides a gentle learning curve, matching you with players that are roughly the same skill level with you, and, on the other hand, provides another measure of progression (summoner levels/exp - you start off as level 1, and after playing many games, end at level 30). Each level brings an extra rune slot and mastery point that you can utilize in game. This would be the "MMO" aspect of the game. It allows you to play your first few hundred games versus other beginners (unlike the BW model of anyone joining your game, and unlike the SC2 model of having only 5? placement games).
3) Marketing
Riot has done an excellent job marketing their game to different markets. They have invested loads of money into the competitive scene in the form of the end of season tournaments as well as partnering with other tournament organizations. There is nothing inherently shady or wrong with getting as much exposure for your game as possible. Every single competitive multiplayer game should be doing the same thing and trying to create a thriving and active competitive scene.
As a sidenote, there is one issue that sc2 players often bring up in threads involving stream viewer numbers (for SC2 vs LoL) that I think is just hilarious. For upcoming and current major tournaments, Riot aggressively advertises it in the game client (something that Blizzard doesn't do). Some disgruntled SC2 players often claim that Riot puts the tournament stream directly into everyone's LoL game client, (which leads to major tournaments often getting 200k viewers). This is incorrect - Riot places a link to the tournament stream in it's client, as well as usually putting a link in the login screen to the tournament stream. If Riot actually had the stream running in everyone's game client, viewer numbers would probably be in the millions, lol. The actual percentage of players that are interested in the esports scene of LoL are relatively small compared to the total amount of players. This is something that Riot needs to address in the future.
As another sidenote, Valve also has news and links in its main menu for DOTA2. If anything, Blizzard needs to adopt a similar approach to get its own casual players more interested in the competitive scene.
In general: LoL is popular because it's F2P model attracts players, while it's casual-friendly balancing, team aspect and matchmaking holds onto players.
Now the second issue: skill.
LoL requires a different skill set than Broodwar/SC2. BW/SC2 are RTS games while DOTA2/LoL are MOBAs. It is impossible to compare skill between the games. MOBAs in general have very limited mechanics. You control only one (main) unit, a couple units at most (due to some champion abilities). In LoL your actions derive from your champion's auto attacks, 4 abilities, any item actives, and your 2 summoner spells. The same applies to DOTA. There are no "buildings" to control and macro from.
However, just because MOBAs are relatively simple mechanically for players with backgrounds in broodwar/sc2/RTS games, doesn't mean the genre is skill-less or has a low skill cap. The emphasis in LoL/DOTA2 is on knowledge over mechanics. BW had a high ceiling for both: you have to be mechanically skilled and have a deep understanding of the game if you want to be the best Broodwar player. SC2 also takes some mechanical skill (obviously not comparable to broodwar) as well as a decent knowledge of the game.
In LoL, mechanics are often simplified to mean last hitting, so as someone said earlier in the thread, a player like Doublelift who has good mechanics, can get a lot of last hits. However, last hitting is only the most "visible" of mechanics in LoL. It is the most easily compared mechanic: if I can kill 100 creeps in 10 minutes, and you only kill 50 creeps, I'm "better" than you mechanically. Mechanics in LoL actually comes down to your champion played - how you use your auto attacks (last hitting, controlling creep waves, or denying in DOTA2), how you use your abilities (what order you use them in, the timing in which you use them, how you use them aggressively or defensively, how you react to other champion abilities, etc etc.) In short, mechanics in LoL derive mostly from your knowledge of the game and knowledge of the champion played than from multitasking speed.
Since mechanically the game is not too demanding, there is not a huge benefit (and really no requirement for) having 300+ apm. Players do tend to spam a lot anyways, but playing with 100-150 apm (mostly mouseclicks) is definitely sufficient.
Skill cap is something that is difficult to judge for LoL. When you watch Flash playing Broodwar, you get the feeling that you could never replicate his level of play even if you played 10 years for 20 hours a day. Watching a MOBA like LoL or DOTA2, even though you see individual players dominating, it is hard to get the same feeling. This is because most of the "skill" is hidden (knowledge skill - simply knowing more about the game, about your champion, etc) and isn't as obvious as straight mechanical skill. The perception that the LoL skill cap is lower than BW/SC2 at the highest levels might be true in a sense, but the game itself is fundamentally different. Because of Riot's policy on balance and champion releases, adjustments to the game are constantly being made. Every couple of weeks Riot releases a new champion or balance patch, often changing the metagame and dominant strategies completely. Players at the top levels must constantly keep abreast of new developments and test out newly released champions (and try to figure them out). Play at the highest levels depends so much on constant changes and new developments, which is why the LoL skillcap hasn't been reached yet - it essentially never will be reached as long as Riot keeps introducing new champions and continues their policy on balance.
(in short: LoL skill cap different - "lower" but constantly evolving - top players must keep up or become irrelevant)
This brings me to my next point regarding practice. Pro teams have to practice a lot. Not only do they need to practice their team communication and coordination (extremely important), but because of the nature of the game, they must constantly keep developing new strategies involving new champions or rebalanced champions. If you think just because LoL is cartoony and "easy" that you (or anyone else - including any Korean BW pro) could pick the game up tomorrow and rise to the top of the game, you're absolutely deluding yourself.
Because of the competitive nature of the game and team dynamics, LoL is no more relaxed than any other competitive game. For solo queue (an essentially random matchmaking system, puts you with 4 other players vs 5 other randoms), passing off losses on the mistakes of teammates may be relatively stress free, but it won't lead to improvement. For competitive play, practice times and the reliance on teammates if anything leads to more stress, not less. If you're playing the game professionally and you're under-performing or slumping, not only are you losing for yourself, but you're losing for your remaining 4 teammates.
I definitely think the main reason why BW pros like Grape see LoL as such an attractive alternative to SC2 is because of the popularity of the game (and subsequent opportunities in the competitive scene), as well as easier "progression" from starting the game to pro level. If they moved to SC2, they would have to start from the bottom, just like everyone else. Unlike Flash, these mid-tier BW pros aren't going to have major sponsorships and opportunities waiting on hand if they switch over to SC2. As opposed to LoL where opportunities are everywhere, the game is only getting more popular, and the game is still relatively undeveloped in terms of professionalism among pro teams. DOTA2 is a different story like some posters have mentioned, as many Chinese and EU teams have already become highly developed professionally due to their many years of playing DOTA1.
Wow this post ended up far longer than I thought. I hope it sheds some light into the issue for some people. I'll probably check this thread more so I can probably answer some more questions if they come up.