Though it will vary by telescope, some of those computers are likely used to control and monitor the telescope. The operator, who may be the astronomer but is usually an observatory staff member, needs to tell the telescope a set of commands to follow -- for instance, where to point and what instruments to use -- and in turn the telescope needs to tell the operator about its status. Some of those computers are used by the operator to submit commands and give remote astronomers access to the telescope, while others give information about conditions at the telescope (wind speed, for example) and diagnostics about the ongoing observation (maybe displaying information about real-time signal processing or local radio frequency interference).
It's possible, particularly for older telescopes, that some of those computers are part of a backend, the second half of the initial signal processing chain, so let's talk about the signal processing that's done in real time during the observation. Say you're an astronomer using a single-dish telescope to monitor a point source. Maybe it's, say, a pulsar. When you point the telescope at the pulsar, what happens to the signal you receive?
- The radio waves arrive at the telescope. Typically, they reflect off a dish and enter the receiver: they're channeled through a feed horn, which funnels them to an antenna. The waves induce changes in the electrical signal from the antenna.
- The signal is probably quite weak, and so it passes through a device called a low-noise amplifier (LNA) to make it stronger.
- In most frequency bands, the signal is at too high a frequency for subsequent processing, so it is "mixed" with input from a device called a local oscillator (LO) to change it to a lower frequency.
These steps usually happen at the telescope; we refer to the above collection of instruments as the frontend. The signal that has now been down-converted to a lower frequency passes through cables to the specified backend. The backend is a second set of instruments that are specialized for the type of science being done. These days, they're usually digital, and may be some of the computers you see in photos.
- The signal may be amplified again, if that was not done before it was transmitted from the telescope.
- Since our backend is probably digital, the signal needs to be digitized, so it goes through an analog-to-digital converter (ADC).
- We usually care about how the signal changes across the frequency band, and so the signal goes through a spectrometer. For pulsar science, for instance, we need to know how the radio waves were dispersed as they traveled through space, and how the pulsar's profile changes with frequency ("frequency evolution"). Therefore, the backend uses a spectrometer to channelize the signal into a bunch of different frequencies across the band.
- Broadly speaking, the data is then written to disk, at which point you can analyze it at a later time.
For radio telescopes with high-quality digital backends, I would expect them to be located beyond the control room. Given the cooling equipment required for the computer racks, the whole assembly can get quite loud, which is distracting for the operator. That's why I'd bet that many of the photos you see just feature the computers used by the operator for control and monitoring.
Regarding your final point: It's possible for you to observe with a radio telescope without being in the control room, yes. Most radio astronomers today using large telescopes observe remotely. It's cheaper to not have to travel. But there is always someone in that control room, the operator I mentioned at the beginning. They need to be able to respond to real-time information on site and talk with other observatory staff members. For instance, maybe it's snowing, and they need to make a judgment call about halting the observation to perform a snow dump to get snow off the dish. It's harder to do that if you're not at the observatory. And if you're using the telescope remotely, it's obviously easy to lose your internet signal; without someone in the control room, you'd be in serious trouble.
(This is partly an excuse for me to give a shoutout to telescope operators, who often work extremely long shifts in an unglamorous job. They know more about the ins and outs and peculiarities of the machines than most!)