Nowadays pop songs are nearly always recorded one layer at a time (the exception being really "old-school" bands). The first track will be a click and/or drum track, then typically bass, keyboards, guitars, brass, strings and finally vocals. Each person recording has to listen to the previous layers, hence the headphones.
String players (and very often woodwind and brass players) always record with the headphones only on one ear so they can hear themselves acoustically in the section. In the phones they will listen to whatever they need to play in time and in tune with the pre-recorded layers. In a situation like this (a pop ballad) they will probably need the keyboards or guitars more than the click.
'Perfect' is a good example of this. If you listen to the finished product you can hear how much had already been recorded before the orchestra got to the studio. That (plus click) is what the musicians are hearing in their phones. The only unusual thing here is recording the lead vocals simultaneously with the orchestra. They presumably had a lot of faith in him to get it right every time.
Note that what you hear and see on the video might not be the take they actually used. You normally don't want a camera crew in the studio making noise and distracting you while you are recording. They probably already had a take they wanted to use and played through the piece one more time just for the cameras.
Even in a situation where there is no pre-recorded or click track headphones are nearly always used, except for classical recordings. In the studio the instruments are very often acoustically separated, so headphones are needed to hear the other players. I've experienced recordings where the orchestra was in one room, the choir in another and vocal soloists in another. That can't be done without headphones.