I’m sure there’s a practical reason, but can’t think of what it could be. Any college radio DJ’s with the straight dope?
I believe it is simply so they can hear how they sound to the audience. It makes sense to me that the earphones serve the same purpose as monitors to a rock band. If you cannot hear yourself that well, it makes it difficult to perform. Even though they are just talking, it is better to hear how you are sounding to the audience. Your voice never sounds the same to you as everyone else because you hear it through the air, AND through your jaw bones and flesh and stuff.
Just a WAG. IANADJ
Not only to hear themselves, but to hear all the other stuff going out over the air. Music, commercials, sound effects, the newsperson in the other booth, their engineer, etc.
Since just playing all this stuff over speakers could cause serious feedback problems with an open mike, you use headsets.
Ugly
The answer to your question is: to hear everyone else. Sometimes there’s a producer in the studio adjacent to them that need to give them directions, comments, etc. and they can provide that through headphones.
The other answer: they don’t. A lot of DJs, even those that play their music really friggin loudly in the studio, don’t wear headphones. The producer can get in contact with them through phones or a speaker while they’re off the air. In fact, most of the DJs I work with I’ve never seen wear headphones 95% of the time.
Having a modicum of experience on this subject…
It’s basically so you can talk into the mike over what’s currently playing without getting gawd-awful feedback.
If you tried to talk over some music and it was playing on the studio speakers, you’d get feedback–the music from the speakers would go into the mike which would be broadcast over the speakers, etc., causing the feedback.
The studio I was in had an automatic cut-off switch. As soon as you punched the mike on, the studio speakers would cut out. If you wanted to hear what was playing (or hear cues from the producer), you had to have headphones on.
Back in the day of vinyl, you’d need to cue up the next track and you’d use the headphones to do this.
You need to do this with tape as well. (Reel-to-reel and 1/8 inch cassettes are still used in some places, believe it or not. 
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(31 years of experience on this one and counting)
When the mike(s) go on the monitors go dead (for the feedback mentioned). Headphones are the only way you
have a clue about what is going on.
They don’t. I have been on the radio playing records a few times and I’ve never worn headphones.
I worked in college radio. You need to do it to hear how you sound over the air. A friend of mine tried to not do it, and he did not know that the music was too loud so you could not hear what he was saying (he was talking over the music.) He quickly learned to use headphones. I would say over 99% of people use them. I see no reason not to use them.
They also seem to need them to hear listeners who phone in and stuff. I was listening to a radio station once and they had a regular spot for a man to talk about sport. He couldn’t get to the studio that day, so he was calling in from a mobile phone. One of the DJs said that he couldn’t hear him, he didn’t have his headphones on. So that’s what I always thought they used them for.
This sounds a lot like my question, but to ask for clarification in this thread. People on TV work without headphones even when talking to callers (talk shows) or with a band playing in the background (SNL, Letterman, etc.). Yet they seem to have no problems with feedback or hearing sound effects, two of the reasons mentioned for headphones.
Several reasons. First, TV studios are (usually) a lot larger than a broadcast booth at a radio station. In the broadcast booth most of the sound is coming from the speakers, which leads to the feedback problem. In the TV studio most of the sound is coming directly from the band or whatever and the speakers there are only a small portion of the ambient noise. (In fact, unless there is a live audience there may not even be a speaker in the studio.)
If you ever see the people behind the cameras, you will notice that most of them are wearing headphones. They are the ones who are hearing what is actually going out over the air. The people on-camera aren’t.
Also, in radio the DJ is usually also running the sound board. In TV, this is being handled by someone in a separate booth. This person is the one handling the mixing to make sure the music isn’t drowning out the announcer and that sort of thing. The on-camera person doesn’t have to worry about this so again they don’t need the headphones.
In radio no one cares what you look like. In TV they don’t want headphones messing up their $500 designer hairstyle. That applies to men and women!
In addition to tanstaffl’s explanation, please note that hosts on news programs and talk shows have a tiny receiver in their ear via which they take instructions from the producer, etc. They are hearing the callers (I believe) through this, not through the studio speakers.
I worked in college radio for many years, including as the general manager at my station, and wearing headphones when on the air was a requirement. I cannot tell you how many times some DJ who thought they were too cool for headphones sounded like a moron on the air, trying to front-announce a song that was potted up too high and drowning them out.