High current means that the amplifier has the capability to supply large amounts of current through the speaker when it is demanded. This can be demanded two ways, either by loud passages in the music, or low impedance in the speaker.
For the same power output, roughly 1.414 times the current is required for a halving of the impedance, and roughly 1.414 times the current is required for the doubling of power output.
So, using an amplifier than can supply large amounts of current is important for the dynamic passages in music as well as being compatible with large numbers of different speakers.
That said, it is important also to be able to supply high voltage levels as well, though this is much easier and cheaper to accomplish in the design of the amplifier.
If you know the peak current delivery of the amplifier and you know the lowest impedance of your speaker, the RMS power delivery into the lowest impedance of the speaker will be:
Peak Current = I
Lowest Impedance = R
RMS Power = (I^2 * R)/1.414
If you know the peak voltage swing of your amplifier, and the highest impedance point of your speaker, the RMS power delivery into the highest impedance point will be:
Peak Voltage = V
Highest Impedance = R
RMS Power = (V^2 / R)/1.414
So, as you can see, there are many factors that go into power ratings and high-current is a necessity as is the proper voltage swing as is just matching the loudspeaker's sensitivity and impedance to the proper type of amplifier, not to mention the 'sound'.
Hope this helps,
Paul Hilgeman