If you're asking for a legitimate explanation for why mathematicians came up with groups, it's because they wanted to find roots of polynomials (or rather, prove one cannot find a general formula for solving large-degree polynomials).
The complex roots of polynomials satisfy symmetry properties. The group structure of those symmetries allows one to discriminate when one can and cannot solve the polynomial using elementary operations (+,-,*,/) and radicals (nth roots). They call this "Galois Theory", and group theory grew out of it to streamline the ideas about symmetry so they could be applied elsewhere, particularly in the study of geometry and non-Euclidean geometry.
If someone's interested in a more detailed discussion of this, here's a an out-of-copyright paper that turned up in a 2 minute literature search: https://www.jstor.org/stable/2972411
The complex roots of polynomials satisfy symmetry properties. The group structure of those symmetries allows one to discriminate when one can and cannot solve the polynomial using elementary operations (+,-,*,/) and radicals (nth roots). They call this "Galois Theory", and group theory grew out of it to streamline the ideas about symmetry so they could be applied elsewhere, particularly in the study of geometry and non-Euclidean geometry.