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In school it was explained through combinatorics, where the factorial among others tells, in how many ways you can order your set. And an empty set has just one, but it has one, order in which it can be arranged. Now that I am writing this, I'm losing confidence, that this explanation is mathematically correct..

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That is the usual explanation, but I don't think that helps much since counting the number of ways to arrange nothing is confusing!

Not as confusing as you'd think, though, if you invoke the rule of product.

The number of ways to do A and then B is the number of ways to do A times the number of ways to do B. The number of ways to do nothing and then A should be the same as the number of ways to do A. This implies that the number of ways to do nothing is the multiplicative identity, one.

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