how can low watt tube amps drive speakers with higher power requirements

It is all based on the quality of the first watt with a lot of amplifiers and speakers and the ability of the amplifier to drive the speaker. The other main factors are room size and listening level wanted. Small amplifiers can be good for a lot of music and speakers but only at low listening levels turn up the volume much at all and you will tax the small amp and you weigh the choice between volume and quality of sound with small amps. The key is to find a good all around amplifier that allows you to play whatever you want on any speaker at any volume, they do exist, but are hard to find.

A low powered tube amp is unlikely to damage a tweeter if driven to distortion, it simply sounds bad.

My favorite  design is a pair of NOS 300B tubes operated in single-ended, ultra-linear mode. About 15 watts RMS output.

Funny!!! 300bs of course can't be operated in ultralinear, on account of lacking a screen grid :)

I read that certain speakers require amplifiers with high "current" and not necessarily high wattage.....and this usually requires a tube amplifier. I believe this has something to do with 20w tube amplifiers driving speakers better than a 150w SS amplifier.

Indeed, wattage is just part of the equation.. Average watts per channel RMS (root mean square) is an average which means there will be peaks and valleys of  power output way beyond and way below the RMS value.  Amplifier manufacturers that do not state power in watts RMS per channel is quite meaningless or hard to understand what they mean.  And the quality of the first watt is super important.  For instance, Klipsch Heritage speaker line with efficiency up in the 98 and above can easily be powered by a 2.3 watt/ch  SET amplifier from Decware (see Steve Guttenburg's YouTube channel on this very topic) if you listen at a relatively reasonable level (vs a night club !).  There is an obsession on the more watts, the better. That is not necessarily true.  PLUS, some speakers have impedances that are all over the map and can approach zero ohms (e.g. almost a short circuit)  and that requires a very well designed amplifiers (e.g. David Berning's)  more than just powerful amplifiers.