I think it is generally useful to try and give a geometric interpretation to whatever we are interested in and groups certainly benefit from such an approach.
Moreover, the groups relevant to physics are very geometric things (they are literally spaces in the sense that the real number line is a space ) and the geometry plays a large role in these groups.
In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play.
Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
Number theory is another example, finding roots of polynomials was already mentioned.
> Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
It connects linera algera to group theory, yes. But representation theory is by far the least "visual" branch of group theory I can think of. It is on a similar "abstract nonsense" level as category theory (and I don't mean that in a bad way, it is just how things are). The only visualizations I can find in my representation theory scripts are commutative diagrams between group homomorphisms, vector space homomorphims and some module endomorphisms and what not :)
Did it? As far as I know, group theory started as method to describe the possible permutations of roots of polynomials, so it was very abstract right from the beginning.
And while there are certain geometric interpretation of some groups and the ways these groups act on objects, the "visual" part stops as soon as the groups gets more general or have more structure.