Well i am pretty sure they are listening to the instrumental while singing, keeping rythm with something you can't hear is rather difficult

Probably a similar reason for life performance, makes it easier to hear the instruments with all the noise coming from the viewers

It's in the name: for *monitoring*.

ie, to clearly hear the rest of the band or the backing track, and themselves.

It's because we like to listen to asmr while we're playing gigs.

Instead of an on stage monitor to hear the band/backing track/click track they use in ear monitors. It's easier to control the volume and the mix.

Most people have covered it, but to elaborate a bit...

In the studio musicians are rarely playing live with the rest of the band, and even if they are playing with the rest of the band they will often be isolated in their own room. The headphones are there so they can hear the rest of the music and/or click track/whatever else they need to hear. This is done because when recording music engineers generally want to get each instrument recorded separately so they can easily manipulate whatever they want on that instrument without causing any issues for the other ones. If, for example, we had a bass and drums in the same room, if one of them made even the tiniest mistake you'd have to re-record both instruments (the bass would be heard in the drum mics, and the drums would be heard in the bass mic(s)). If the instruments are separate then you could just re-record whatever little part you needed to. This also applies to any other treatment the engineer might do, compression, EQ, pitch/timing adjustments..etc. Its basically always better to have the instruments separate for most modern recordings. Things like orchestras often record as a group because the interaction of that many instruments in a room sounds different than just playing each instrument separately and playing them back at the same time, and most classical recordings aren't going for the same level of inhuman perfection that most pop/rock/other modern genres are.

For live performances it's because of a few things, but mostly 1) stage monitors kinda suck, and 2) stages are often really loud. In ear monitors allow musicians to hear everything they need more clearly and at a significantly lower volume than without.

Both have to do with being able to hear what you're doing.

The singer needs to hear the track in order to sing to it when recording. If you use anything other than headphones you'll pick that up in the vocal mic.

Musicians moved to in-ear monitors because hearing everything through stage monitors kind of sucks. The speed of sound is really slow so you're hearing things coming from all over the place not perfectly in sync. In-ear monitors fixes this, the whole band gets the perfect monitor mix wherever they are on stage.

They do it to fuck with you, OP

There are different reasons, sometimes is so they can listen their own voice, other times it includes the tick of a metronome (to keep the pitch or pace of the song), sometimes even to hear the next lines, etc. But normally is to hear their on voice, coordination with the rest of the band, and the pitch.

Headphones? I mean they mostly use In Ear Monitors now but it's mostly to track a click track because listening to the speakers in the room can fuck your timing up. reverb, echos and all

As others have pointed out, we do it because it lets us hear things better. What I haven't seen pointed out (could have missed it) is that as musicians we generally don't hear what you think we hear. Either in the studio or on stage, each of us wants a unique mix. For example, I play keys (a percussive instrument) so I generally want more drums and bass, with less rhythm guitar and even less lead guitar. I want to be able to hear them for cues, but they're super quiet in my mix. The only way to do this for every musician is for every musician to ONLY be able to hear their own mix, so you use headphones (studio) or IEM's (stage). If the audience heard what I heard, they'd probably think the band sounded awful, but that's what I need to be tight - especially during improvisation or complex segments that require syncopation / doubling / whatever other fancy stuff my guitar player is addicted to that day.

The alternative, if you're wondering, is that everyone's mix on stage bleeds into the other mixes - which leads to a lot of noise competition as the night goes on and hearing goes out.

This video goes into a lot more depth about monitor setups, but it definitely answers the question on why musicians wear headphones https://youtu.be/mHoljbkyAEs

Real time monitoring.

I'm always intrigued by this because I can't even take phone calls through IEMs (as I can't hear myself speak and it's very distracting). So I presume the musician can hear everything including their own voice without any noticeable delay?