Yeah, what a lot of people forget is that guitar playing isn't just hitting notes.
It's also learning to mute unplayed notes with the left and right hands.
Over a period of years, it becomes second nature but for beginners it's hell at first. A couple of tricks you can use is to back off on the guitar volume and treble a bit. (most guitar players seem to like playing at full volume and treble for some reason).
You can cut back a little on the overdrive if you're using it.
You can also use a lot of right hand palm muting over the bridge during leads.

Other than that, it's a whole bunch of tecniques that you learn over time.
Just bending one note up and down on the 3rd string, 7th fret for instance can include using your left hand thumb and third finger to mute the other strings while you bend up, then as you're bending back down using the first and second fingers of your right hand to block the 1st and 2nd strings to stop them from ringing on. I'm constantly using fingers that aren't actually playing anything to mute strings.
Of course in the studio you can always cheat a little too.
If you're playing a lead that only uses the 3rd and 4th strings, take the other strings off... or use a hair braid to mute all the open notes or use a piece of duct tape over the 5th and 6th strings if you're playing D chords on an acoustic.