I'm going to shoot for a non-technical explanation which is what I think you are looking for.
We'll take two examples. #1 is a speaker that has a nice flat impedance of 8 ohms across its entire frequency band. Devore speakers, for example, are designed with this goal in mind. This type of speaker is easy for an amp to drive. Any good quality amp that is rated for enough watts to make the speaker go loud will allow the speakers to operate with a reasonably flat frequency response.
Example #2 is a speaker that has a variable impedance, say 8 ohms, at some points in the frequency spectrum and a lower impedance of say, 4 ohms, at other points. When the impedance drops the speaker will draw more power from the amp if it is available. If it is drawing 20 watts at 8 ohms it will potentially draw 40 watts at 4 ohms.
So what happens in example #2 if the amp is rated for 100 watts at 8 ohms and 100 watts at 4 ohms? The speaker will not produce a flat frequency response. The amp doesn't have enough current to double its power into 4 ohms and the result will be reflected in lower output in the frequency range where the impedance is 4 ohms. In other words, using a low-current amp on a demanding speaker acts like an unwanted tone control.
So why don't all amp manufacturers make their gear so that it doubles the output with each halving of the impedance? The simple answer is cost and practicality. I'm running a pair of Thiel CS6 speakers which dip below 3 ohms in part of the spectrum and I'm powering them with a Krell KSA 300S which puts out 300 watts at 8 ohms, 600 watts at 4 ohms, and 1200 watts at 2 ohms. This amplifier was quite expensive in its day (mid 90's), weighs 185 lbs, and runs very hot. Most people would not consider it practical to own.
So why don't all speaker manufacturers make their speakers with a flat impedance curve? There are hundreds of tradeoffs in speaker design and several high end speakers have difficult impedance curves which require monster amps. Wilson and Thiel are notorious examples. In order to meet their specific goals for sound, these speaker designers are willing to require their customers to buy a very limited selection of amps that can supply high current (a lot of watts into low impedances). At audio shows you will often see large Wilson speakers powered by D'Agostino amps which are heavy, expensive, and double their output with each halving of impedance. The large Pass amps also have this capability.
A clarification: Impedance is different than sensitivity. There are speakers that have low sensitivity (say 86 db) that also have a nice flat impedance curve that stays around 8 ohms. In this case you can get away with an amp that puts out reasonably high wattage (say 200 watts) but doesn't increase its output at lower impedances. Even with the high wattage an amp of this type can be reasonably small and light weight or be powered by tubes. The most brutal load that an amp has to deal with is a low sensitivity speaker that also has a difficult impedance curve. This is what people mean when they say that certain speakers "need a lot of current."
Because there is such a range of sensitivity and impedance between speaker models it is imperative that a buyer analyze how much power the speaker needs before buying. If someone bought my CS6s and hooked them up to a high quality expensive tube amp they would sound terrible. There would be a nasty suckout in the upper bass region and the sound would be anemic. On the other hand, you wouldn't want to hook up my big Krell to a pair of Avantgarde horn speakers that only require a few watts to play loud.
So the point of all this is "current" has become sort of a general way to warn potential buyers that certain speakers need a beefy amp. The specification dealing with output at various ohms is critical to note. Matching a speaker to an appropriate amp is a big factor in achieving good sound.
Here is a good video from John Devore that explains this stuff in basic terms.