Speakers are current driven complex impedance devices. They are not a purely resistive load. Because of that complex impedance speakers require more current at some frequencies tha others.Ohms law applies, but only at a given frequency will the calculations be accurate.
Amplifiers ability to deliver current to loudspeakers is gated by the current carrying capacity of the output stage transistors or tubes and what the power supply can deliver.. Bigger power supply caps simply give more reserve under demanding loads within the limits of the output devices.
To protect the output stage, some manufacturers implement current limiting that prevents the amp from trying to deliver more current than the output stage can safely deliver. This happens especially at low frequencies and high volumes, but the audible impacts are throughout the spectrum.
Power supplies, especially the capacitors, and the output transistors are the most expensive components in an amp. This why 'watts per dollar' is a terrible metric it inevitably leads to amps that measure well on the bench, but donr sound very good, especially when pushed, or when driving demanding speakers.
Finally speakers generate their own current, called 'back EMF' when a coil moves in a magnetic field. This has to be absorbed and dissipated also by the output stage of the amp, so more load on the output to be accounted for. This also where the output impedance of the amp comes into play, as that determines the damping factor.
Bottom line, high current amps are in general, a good thing, but not the only thing