I was trying to explain to a non-pilot why increased power is required
with
altitude. She said "isn't the air thinner up there so there isn't as
much
resistance?" I said "yes, but the plane needs to fly fast enough for the
air
over the wings to feel like it does down low. So the speed required goes
up
you get higher. More speed need more power."

This didn't really do the trick.

Can someone think of a better way of putting it without resorting to
mathematics and an explanation of IAS and TAS?

TAS increases with altitude for a given power setting due to less
aerodynamic drag at higher altitudes. It does not take more power to go the
same speed at higher altitudes - at least, not in any of the airplanes I've
ever flown. Take a look at the speed/power charts for a turbo and you'll
see what I mean - if you maintain 75% power the higher you go the faster you
go.

If you're talking about altitude effects on the power output of a
normally-aspirated engine, that's a different story. At about 8,000 feet a
normally-aspirated engine will probably be putting out around 75% power at
full throttle, and it will continue to decrease as you go higher.

BDS