Let me add two other guesses:
Especially in a class and/or for students first encountering this material in a book, it is best to start with concrete representative cases. $\Gamma(N)$, $\Gamma_0(N)$, and $\Gamma_1(N)$ certainly qualify as concrete, while also being sufficiently general to illustrate many of the fundamental ideas.
$\Gamma(N)$, $\Gamma_0(N)$, and $\Gamma_1(N)$ correspond to three extremely natural answers to the question: "What sort of level structure should I add to the moduli space of elliptic curves?" Very early in the study of elliptic curves, one sees the importance of the torsion subgroup. So it makes sense to classify an elliptic curve with a cyclic subgroup of order $N$, or a point of order $N$, or all of its points of order $N$. (Yes, I know $\Gamma(N)$ doesn't quite do the latter, but close enough.)
