We do this for a couple of reasons. Secondly, if we do all of our amplifying at the frequency which we are receiving, there is a good chance that some of the amplified signal will escape back into our antenna and produce feedback. Its the same problem you have when you get the microphone too near a public address speaker. We pick the lower of the two outputs by passing the mixer output through a filter in the IF amplifier. The local oscillator produces a signal which is injected into the mixer along with the signal from the antenna in order to effectively change the antenna signal to a frequency which can be handled by the IF amplifier.
Many radio telescopes use a quartz crystal derived local oscillator signal.
Quartz crystal oscillators are quite stable and drift little in frequency. Because most radio telescopes are quite broadband in nature, a small amount of frequency drift in the local oscillator may be tolerable. One must be careful, however, that the drift is not so great that received signal frequency begins to infringe upon the bandpass of the antenna or any front end filters that precede the mixer. Also, it is possible to drift into an interfering signals frequency. When either of these things happen, it can result in a change in the output of the telescope which could be mis-interpreted as a real change in received cosmic noise.