It's not about optimal matches at all - it's about when to stop looking.

The assumption is you don't known the set of potential matches, or the order they come in, or anything really. But there is a deadline for the decision (or a maximum number of attempts). So how to balance making attempts to gather information with committing to a final decision so you don't run out of time? All else being equal, the rule is 1/e. Spend the first 37% of your time/attempts gathering information, then commit to the next option that's better than you've seen.

This doesn't guarantee a good match (or even a match!) but probabilistically the strategy is optimal.