The output of the mixer tube is actually several frequencies, most importantly the base frequency minus the local oscillator frequency, the base frequency plus the local oscillator frequency, and the base frequency alone. The IF transformer is an LC circuit tuned to resonate at just one of the above two differential frequencies, the one which gives 455 kHz (or whatever the IF frequency is designed to be). This allows the IF transformer to pass the 455 kHz signal on to the next stage for further amplification or detection, while suppressing the other mixing product frequencies (since they are not at the resonant frequency of the IF transformer).
Daniel:
Let me add to CV's answer. In the typical AA5 radio, you will have three resonant cirucits: One, at the antenna, you resonate the incoming frequency (usually the loop antenna) with the large section of the tuning capacitor. This is done to attenuate the image (I'll get to that later). The second resonant circuit is the small section of the tuning capacitor and the oscillator coil. The local oscilolator operates at a frequency 455kHz above the incoming frequency. The mixer stage (usually the local oscillator and the incoming signal use different grids of the same tube). There are four frequencies coming out of the mixer. The incoming signal, the incoming signal plus the local oscillator (The intermediate frequency), the incoming signal minus the local oscillator (the image frequency I mentioned), and the local oscillator. We now go to the IF amplifier, which is our third resonant circuit (so it is four tuned circuits) and that strips away the incoming signal and the local oscillator. A frequency of 910 kHz above the incoming frequency will also come out of the mixer as a frequency of 455, so once again, that is why we have a resonant circuit at the antenna.
By the way, aircraft VHF navigation is between 108 and 118 mHz, and the local oscillator of a FM radio (with an IF of 10.7mHz) will operate in that band, so that is what started operating electronic devices on aircraft a no-no.
Lewis