[quote=@Clever Hans]
Does rolling two of the same die and taking the higher result actually give you better odds of producing a higher result?  This is what's really making my head spin.
[/quote]

Short answer: Rolling multiple die and taking the better result not only more frequently yields a higher result, it more consistently avoids lower rolls.

In the game you describe, you roll 2 dice, and you always take the die with the better result. If you were just rolling 1, you have a 1-out-of-6 chance to get a 6. This is easy to see, because all we need to do is count the number of sides on the dice, which will give us all our possible outcomes.

1,2,3,4,5,6

It gets trickier for two dice. Because to find every possible combination, we need to multiply this by the number of sides for each additional dice. 6 X 6 = 36, and all the combinations are as follows:

1/1,1/2,1/3,1/4,1/5,1/6,
2/1,2/2,2/3,2/4,2/5,2/6,
3/1,3/2,3/3,3/4,3/5,3/6,
4/1,4/2,4/3,4/4,4/5,4/6,
5/1,5/2,5/3,5/4,5/5,5/6,
6/1,6/2,6/3,6/4,6/5,6/6,

See? 36. If we rolled 3 6 sided die, we'd have 6 X 6 X 6 = 216 possible combinations. But we won't worry about that for now.

So going back to our possible outcomes for a single die, you can see that there's a 1-in-6 chance of getting a 1 or a 6. But when using 2 dice, it's clear things aren't quite so balanced. Statistically speaking, you have a 1-in-36 chance to get a 1 and a gnarly 11-in-36 chance to get a 6. Had you been rolling a single die, you'd roll a 6 6 times, almost half as frequently. And you'd have rolled 5 more 1s. 

EDIT: should probably mention that the number of outcomes does not always equal the percent chance of getting a particular outcome. But in this case where all the dice have the same number of sides, each dice has the same % chance to land on any particular side.