는데

Gives background, then trails off

Sets up background and leaves the door open for what comes next: "the enemy is outside — so why are we fighting?", "I met a friend the other day, and...". Sometimes the second half contrasts with the first, sometimes it just needs the context, and very often the speaker stops right there and lets you infer the rest, which is why so many subtitle lines simply end on it. Add 요 for a polite version. Compare 지만, which is a flat "but" and does nothing else.

Register: very common in conversation