how does current work in an amplifier?

A high current amplifier has a very large transformer with a compliment of large capacitors. A SS amp has an array of semiconductors (transistors or mosfets) that are either cascaded or cascoded. These devices are connected to the transformer and capacitors and function as variable resistors to create a large, powerful signal that is identical to the small input signal coming in from the preamp. That large, powerful signal drives the speakers. The Capacitors and Transformer are energy storage devices. They provide the peak power needed as the music signal varies from low to high. Imagine having a big rheostat that ranges from -60 to +60 volts that is connected to a speaker. If you could turn the dial back and forth fast enough to mimic a musical signal- you could make music. A tube amp functions much the same way. The tubes get the preamp signal and vary their high level output voltage to mimic the musical signal. The output of most, but not all tube amps goes through transformers. That limits their current carrying capability and that is why tube amps generally have the same power rating for 16, 8 and 4 ohms. SS amps can output higher power levels into lower impedance speakers up to the point that the semiconductor devices overheat and go up in smoke. To demonstrate, apply a dead short across your SS amplifier. (Although many modern SS amps have overload protection devices which kind of takes the fun away). Very simplistic terms, but that is how any amplifier works.

PS. I should mention that applying a dead short for smoking affect will require extensive repair of your amplifier. Also, a dead short will not hurt a transformer coupled tube amplifier, but would not be good for the tubes in an OTL version.

So.....all other things being equal (same room, same source, same cables, same other characteristics of the amplifier), am I to understand that an amp that delivers more current would do a better job of driving a 4 ohm speaker, like Magnepan MG12's, than would an amp with less current?

Actually, as 4 ohm speakers go, I think Magnepans are less current hungry than many. They do like lots of power though.

It just depends on the amp-speaker combination and how high a listening level you prefer. Once the power supply of the amplifier is maxed out, the output signal voltage is not going to be able to match the input signal waveform, ie. distortion. The bass requires the most power, so it is the bass that starts to sound flabby or weak. At least, that is my experience...