Does this just use the "trade value" from FBG or does it calculate a player value based on the scoring system?

Based on the your scoring system.

More generally, how does it work? I must be missing something, 'cause I can't figure out how to move somebody from one team to another in the Trade Analyzer.

Thanks

With your team in the trade window, remove the player(s) you're giving up, and add the players you're getting. Voila!

You can re-add players from before to after using the center arrows.

If you care about the other team's outcome (I don't particularly), just repeat the process on their team screen.