In short? Those are in ear monitors.
A more detailed explanation of monitors and their use in live performance:
You may recall seeing wedge shaped monitors on the floor at the front of the stage at concerts? Those are monitor speakers. If you have the equipment and personnel, a different mix goes to those speakers than the one that goes out to the audience (and a different mix might be going to tape or the air).
You can see floor monitors in this photo of a Yes concert:
Why all these mixes? Why not just pipe the same mix through?
The band doesn't need to hear everything. The more sound there is up on stage, the more difficult it can be to control feedback or for the vocalist to hear themselves (which is critical for keeping time and pitch), so the on stage monitor mix is usually stripped out of everything but vocals and some key instruments that they might use as a reference. Every band has its preferences, so some have a fuller monitor mix.
The house mix would be what goes out to the audience. It might be different than a mix going to tape because it is adjusted to reflect the acoustics of the venue. Without that adjustment, you can focus on just what's coming into the mix, without having to equalize or adjust levels based on how it sounds coming out of the house speakers and bouncing off bodies, concrete, etc.
The wireless earphones are essentially replacements for those wedge shaped speakers. They have several advantages over a floor speaker. Since it's stuck in your ear, feedback is much less likely. Because others can't hear what is in YOUR earphone, each band member can have a specially tailored mix. Bass and Drums are an important element in music, so the bassist and drummer, might have each other in the mix at a higher level, to help keep themselves "in the groove." A vocalist might have the other vocalists at a higher level in the mix, especially if there is any harmony. Plus, you can move anywhere on stage and still hear YOUR monitor mix.
In a television situation, they could also have a producer's mic routed to their in ear mix. If a musician isn't looking up enough or strays away from a key camera spot for a solo coming up (for a zoom in), it could serve as a means to communicate that.
On talk shows and news shows, the on air staff usually wear in ear phones so their producer can communicate with them. Typically such things as, "30 seconds to commercial", "pick up the pace", "expect a breaking story any minute", "we're working on it now, stuff like that." In the old days, allowing that to be noticed was considered tacky or a no no. These days, the on air person will often comment that "they'd just been told..." or if fussing with an earpiece that won't stay in.
In the second or third line of the song, the lead vocalist reaches down to his waist briefly while at the piano. He was probably turning his monitor level up. No need to point at a speaker and look to the guy running the monitor mix off stage and motion upwards... The in ear monitors are more expensive than running a speaker on stage, but they're much more flexible. Of course, they can suffer from the reliability issues of wireless signals...