Why does the frequency of carrier signal not matter when we impose the original signal on it?
The frequency of the carrier does matter. You need to pick one that's suitable for your antenna, and doesn't interfere with, or get interference from, neighbouring services.
However, at the receiver, when you demodulate the modulated carrier with respect to the specific frequency you chose at the transmitter, you remove the effect of the initial choice. You could choose to transmit with 100 MHz or 500 MHz carrier, and as long as you convert that signal down with the equivalent of a 100 MHz or 500 MHz reference, the two systems will function the same.