Explaining to laypeople why bootstrapping works

I recently used bootstrapping to estimate confidence intervals for a project. Someone who doesn't know much about statistics recently asked me to explain why bootstrapping works, i.e., why is it that resampling the same sample over and over gives good results. I realized that although I'd spent a lot of time understanding how to use it, I don't really understand why bootstrapping works. Specifically: if we are resampling from our sample, how is it that we are learning something about the population rather than only about the sample? There seems to be a leap there which is somewhat counter-intuitive. I have found a few answers to this question here which I half-understand. Particularly this one. I am a "consumer" of statistics, not a statistician, and I work with people who know much less about statistics than I do. So, can someone explain, with a minimum of references to theorems, etc., the basic reasoning behind the bootstrap? That is, if you had to explain it to your neighbor, what would you say?