Amplifier current vrs watts; why is current more important?

Because amplifiers don’t deliver watts. (How many watts of light does a lamp deliver? Depends on the bulb, i.e., the load).

What they mean is that amplifier wattage is rated at load of 8 ohms and any change in that load will demand more current from the amplifier.

Example:

100 watts into 8 ohms ---> Voltage is 28V, current is 3.5 amps, transformer 100 VA (~watts)

When the speaker dips to 4 ohms, the voltage can only remain constant if the power supply can deliver 7 amps. To do this, the transformer has to be at least 200 VA. Twice as big.

At 2 ohm load, the PS has to supply 14 amps to maintain voltage, meaning the transformer has to be 400 VA minimum.

If the voltage cannot remain constant -- the transformer/filter caps too small -- then the amp has a hard time driving the speaker through its varying load. This is an oversimplification but by Ohm’s law shows why current is dependent on the power supply and not the nameplate rating.

Also a read from Pass Labs First Watt will help.  

Current is only of import when speakers dip to low impedance or have challenging phase angles, or both. Otherwise it is of little consequence.

There are so many wrong's in this statement.

Your typical amplifier outputs volts

It may be a bit oversimplification.  Two power amplifiers that have the same power supply voltage may not have the same output current.  Ultimately it's neither the voltage or current, but the overall design that determines the quality of the sound.

You may have a massive 800W power amplifier with massive output current but it may not provide quality music reproduction compared to a more modest 250W amp.