In amplitude modulation is the carrier wave multiplied by the radio signal wave to form the equation of the resultant wave to be transmitted?
It appears remarkable to me that electrical signals would multiply
together

Think of an audio 1 kHz tone going through an amplifier to a loudspeaker. Then, when you adjust the volume control up or down, the speaker output gets louder or softer. That is precisely what multiplication is in AM broadcast transmissions. The 1 kHz tone represents the carrier wave and, the relatively slower changes in the movement of the volume control represent the modulation signal.
And, in reality, that's pretty much what a broadcast AM transmitter uses - a voltage controlled amplifier where the control voltage is the modulating signal and the input is the carrier frequency. That is multiplication (2 quadrant type).
There are more esoteric types of modulation but grasp this first and you are well on your way.