To trigger romance with Carth, Bastila or Juhani, speak to them often, especially after you finish each planet, until the topic is exhausted. Try to be sympathetic with them, otherwise the romance may end at some point.
You must complete their Side-Quests for the subplot to continue.
1) Bastila (Male only)
Part 1: Kiss with Bastila
Bastila decides to give in to her emotions, and asks you to kiss her
Bastila tempts you to the Dark Side, and professes her love for you
You must trigger the romance before the Leviathan. If you don't have a kiss scene (or her confessing her love for you) by then, the plot will end.
Once you have a kiss scene, don't speak to her about it! There's a bug which will kill the romance. You can confirm her love on the Temple Summit if you choose the right option as shown in Part 2.
2) Carth (Female only)
Part 1: Future with Carth
You and Carth talk about the future and love
After you resisted the Dark Side, Carth & you profess love for each other
[Watch out for...]
After the Leviathan, talk to Carth until he opens up about you being Revan. If the scene in Part 1 doesn't occur, you didn't trigger the romance with him.
If you choose the Dark Side path on the Temple Summit, the scene in Part 2 will be replaced by the Dark Side outcome. If you have already triggered the scene in Part 1, you'll have to kill him on the Star Forge after defeating Malak.
(This section will be updated once I make videos)