This is my attempt to synthesize my ideas about getting good at Brood War into a single thread.

I think there are three key skills:
1) A vision
You need a vision of the well-executed siege of your opponent towards which you are building.
You want to consider how every unit or combination of units can be a tool in your tool-belt for arriving at and winning a siege.
Where a tool is lacking, consider all the ways to pair it with other tools which help cover its deficiencies - even if merely by berserking harder so the opponent doesn't have time to create some monstrosity that takes out your first tool.
Of course, it is important not to get run over by an enemy army that is "just stronger" and bigger.
Consequently, you start with whatever you need for the early game and midgame and add in tools to build it into your lethal siege.

2) A sharp eye
My advice is to look for situations that likely indicate weakness by the opponent.
When Protoss shows only 1 Zealot 1 Goon vs Terran for example, the Terran needs to consider this weakness might be real and attack.
In general, you want to attack any sign of underlying weakness early on in the game and early on in your training.
Eventually you will fall into some traps, and may have to randomize whether or not you attack even from the same scouting information.
Playing very aggressively early on will help you map out all of these early attacking options.
Understanding the early game very comprehensively will allow you to avoid the opponent being able to get away with too little army / too much weakness / too much growth / too much econ or tech.
This will save you from entire worlds of nightmares.
You also want to play to cover a wide range of possible moves.
You'd like to cheaply poke, attack, scout, etc so that even if that move fails you're likely winning over time.
Often, these moves provoke a defense that is ultimately more costly than the move itself, and the attack is justified even when responded to properly.
You can't leave these early-game chips on the table and play at the highest level.
It's also fun because you get to win/lose games quickly and you learn a lot.

3) A mirror
As you get better and better at spotting weaknesses that need to be pounced on while developing towards a full army, look at your own play by starting with Opponent POV in replay.
Where are all the weak spots that could have hurt you?
Where are the weak spots in the opponent that you should consider exploiting?

My BroodWar ID is Bushido