There are a lot of really good discussions online already, but what it boils down to is that the string doesn’t only vibrate (oscillate) along its full length (the fundamental)—it also vibrates as if it was three strings each having a third the length, and also as if it was five strings each having a fifth the length, and so on and so on.
The frequency of an oscillating string is inversely related to its length (other things being equal, namely density and tension). So a shorter string will oscillate faster and produce a higher frequency than the fundamental.
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