From my knowledge of the legal field - not personal but through a family member - I believe their "internship" as such is a made-up thing. Matt and Foggy appear to be at the point as working as summer associates (like an internship, but not called that). The firm then hires its favorite summer associates to be full associates, and they eventually work their way up to partner from there. Wooing summer associates is a very real thing, and who knows how extravagant that could get for a big New York firm.

Which doesn't actually answer your question, but my point is that I don't think the show is particularly invested in portraying anything akin to accurate law school/young associate experiences, so your standards of realism aren't too high to start with.

Possible reasons for flying?
-attending a meeting/conference (shadowing a partner at said meeting? I find it hard to believe the firm would ship them off on their own)
-visiting one of the firm's offices in another city, if they have multiple locations
-meeting a major out-of-state client