I think the idea is, say the average person uses 100 gallons of petrol a year. You add a $1 tax to each gallon, thereby costing the average person $100 per year, but you give them $100 back. The average person is no better or worse off, but it's an incentive to use less fuel.

You could make the argument either way that this fall's more on the rich. They tand to have more bigger cars, bit then you could argue that the poor have older less efficient cars, but it isn't really aimed at wealth. I suppose the key is that any party can arrange their lifestyle to avoid being worse off.

Why would significantly less money be taken from the wealthy?

You set the marginal rates, or the parameters of the function for determining the marginal taxes equal to however much is needed to cover the costs.