When I touch the probe I see a 50Hz sine wave with an amplitude of about 100V from peak to peak. I understand that 50Hz is the mains hum picked up through my body, which acts as an antenna, but where is that huge amplitude coming from?
Your body is only very "loosely" earthed. In fact there could be tens or hundreds of Mohms between your body and earth.
This is one of the reasons why the body can accumulate charge (through movement) and then you feel the discharge when you touch something or someone that is earthed.
The cables in (and around) your house produce an electric field that is 230V (or whatever your AC power live voltage is) close to the cable and 0V at earth. If your body is "somewhere" between live and earth, the voltage that will appear on your body is also going to be somewhere between live and earth.
The oscilloscope usually measures a voltage relative to earth and it can have several Mohms input impedance. This means when you touch the probe tip the screen shows a large voltage is present. Because the signal on the scope is AC, the capacitance of your body to earth and "live" also play a significant part in your scope showing this signal.