How this circuit actually modulates the carrier with the input (message) signal
The diode is used as a blocker so, if the modulating signal is lower in amplitude than the carrier signal then it limits the carrier amplitude. In limiting the carrier amplitude it imposes the shape of the modulating signal onto the carrier envelope on the left of the 10 kΩ resistor. The trick is to keep the modulating signal peak less than the carrier peak signal so that there is always carrier envelope shaping occurring.
This is amplitude modulation but, it also contains the modulating signal as well. That modulating signal is removed by the band-pass filter leaving the modulated carrier at the output.
Which part of the circuit causes the carrier amplitude to vary
The diode is the crucial part of the circuit.
Why the peak-to-peak of the modulated signal is 1.25V instead of the messages 2.5V
I think you mean "modulating" signal rather than "modulated" signal. The modulated signal is the modulated carrier so I think you will agree.
Anyway, the circuit is so very simple so you have to get amplitude levels in the right area for this to work without excessive distortion. For instance, in my original circuit I used DC offsets to tweak the modulation process: -
It wasn't a definitely thought-out design; I just wanted to show how really clean AM can be obtained with a low component count and little circuit sophistication.
You might get more insight by using a buffer (as per my original circuit) and looking at the waveforms on the input to said buffer.