As we know from Galois theory, an irreducible polynomial is soluble in radicals if and only if its Galois group is solvable. However, solvable groups seem to have an importance in group theory far beyond their implications for polynomial equations. For example, much effort was expended on proving the Feit–Thompson theorem, which is one of the pieces of the classification theorem, but only its corollary, that all finite simple groups of odd order are cyclic, is required for the classification, and perhaps (I do not know) this could have been proven without using the notion of solvability.
Why are solvable groups such an important subset of groups that so much research has been dedicated to their properties?
I trust you'll grant that abelian groups are important and deserve plenty of research.
Abelian groups have these properties (among others):
Subgroups of abelian groups are abelian,
Quotient groups of abelian groups are abelian.
But if $N$ is normal in $G$, and both $N$ and $G/N$ are abelian, that doesn't guarantee $G$ is abelian.
Solvable groups have these properties:
Subgroups of solvable groups are solvable,
Quotient groups of solvable groups are solvable,
If $N$ is normal in $G$, and both $N$ and $G/N$ are solvable, then $G$ is solvable.
In a sense, solvability is inherited, both going down and going up, so it's a nicer property than commutativity.