I heard the phrase "write from the subconscious" used when discussing a technique in a writing course and it perked my interest. What exactly does it mean?

I think this means writing without a plan or outline. Just write the story as it comes to your imagination at the moment.

I used to do this when I first began writing. It is fun, relaxing, and very satisfying.

Essentially, you know your characters, and the situation they are in. So, as you're writing, you ask yourself "what would he say now?", or "how can she get out of this mess?", and then you write it.
And you don't know how the story will end until you get there. That's what makes it fun.

You'll need a natural storytelling ability for this, and be able to write nearly as fast as you can imagine things.

I don't write this way, these days, because I'm trying to complete a planned work. I know the beginning and the end of the story.