Mastering can involve making a new transfer from original tapes (in hopes of getting a better starting point than the last transfer), but mainly it is sonic enhancements: noise reduction, EQ, and dynamic compression/expansion. Judicious use of these tools (not just turning all knobs to "11") can result in a more polished, modern sound that makes each song sound as good as it can (in the engineer's opinion); and it can smooth out inconsistencies from song to song (so everything on an album sounds like it belongs together). *Re-*mastering is just when someone does all this work again, often just for the sake of being able to market the same music again as "improved".

The ELI5 version is more like:

You know how your audio player has an "equalizer" or "tone controls" where you can change the volume of, say, just the bass and treble pitches in the music? When you mess with those settings, you are, in effect, "remastering" the audio as it plays. Well, someone did that (and a few other things) on every piece of recorded music you've ever heard, after it was created, to make sure it sounded really "good", in their opinion. This was "mastering" the audio. At some point, when the record company wanted to sell more copies, they got someone else to do that work again, and they marketed the result as "remastered", implying it sounds even better now. Whether it sounds different, and whether it is actually better, is often a matter of taste.

Source: I have done mixing and remastering work professionally.