They don't represent higher testosterone levels. Nor a better chance of better offspring.
Indirectly, with a height advantage, all other things being equal, they're more apt to win a fight. Women like more "masculine" men. Much like you see many gals going ga-ga over men in military uniforms, being cops or firefighters, or driving a big truck ("get er done!"). That's one avenue that "manliness" can hit home to a gal. But back to the central topic...
If it's just biologically about being more apt to win a fight (historically guys' bodies more or less the same compared to today; no gym) -- wouldn't a ripped, muscular 5'6" guy be wanted by a 5'3" gal moreso than a lanky 6'0" guy? He's still taller than her... and let's assume both are equal on everything else and have at least cute faces...
Well, with their shirts off it will close the gap, sure... but even though that 5'6" guy would kick the 6'0" guys' arse, why does it still stand? Because it's a more deeper biological thing, that's why.
Some girls aren't so into that -- and as long as the shorter guy is outwardly masculine/built than the other guys in the environment, the gap closes quite well for many. BUT, as you pointed out: He DOES have to be taller than her IF she's not a notably tall woman (then it gets more complicated).
If you're 5'5", yeah, you're going to have issues about that. I'd get shoe-lifts that have some heel on them. No, they don't compare to chicks in heels. :) But they give you 1.5-2 inches more than regular shoes, depending on the variance -- and when SHE puts on her heels, it will lessen the blow, so you can still at least date a girl you're not much taller than.
In essence: Most girls, notably social & attractive ones, want a guy who's taller than the average guy in the room. That's what it boils down to. Many gals notices it visually, and it enhances their vision of "more" of that man (=a sense of more masculinity with biological desire roots).