だって means “but… / because…”. At the start of a line, だって is an excuse: “but…”, “well, because…”. だって眠いんだもん = “but I’m sleepy!”.
Is だって rude? Who can you say it to?
With friends: OK. With coworkers: careful. With a boss or teacher: avoid. Casual. It can sound defensive, like making an excuse; in a meeting, use というのも instead.
How common is だって in real speech?
In 1,905 live comedy routines, だって appears in 362 lines from 129 comedians.