The title says it all, but I want to frame the issue two ways, in hopes of a slightly more in-depth answer.
With Bitcoin, the blockchain is transparent. In the "ordinary course of business" in the world of Bitcoin, it's rather simple for miners to quickly prove whether or not an output is available to be spent. But how does it work with Monero?
What made me think to ask this is that when recovering a wallet, it took my computer hours to scan the blockchain, presumably using my private send key and private view key all along the way, while it determined which outputs were in my address and which of those outputs I had spent as inputs in new 'send' transactions. If my computer has to work so hard to prove that it has legitimate outputs to spend, how can blocks be mined so quickly (in minutes) without double-spends sneaking by?