I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of automorphisms on $X$.
So now comes algebraic geometry. Here the geometric data is in the form of sheafs of functions on the space $X$. But is there a canonical way to get a group of transformations or an inner product out of this data, i.e. is there any connection to "Klein's geometry"?
You can interpret Klein's point of view on geometry very, very broadly as being the following: geometry is dictated by the morphisms you choose between geometric objects. In the usual interpretation "morphism" is taken to mean "isomorphism," but from the point of view of category theory there's no reason to make this restriction. In other words, geometry is dictated by the choice of a category of geometric objects.
The geometry in algebraic geometry is dictated by the fact that the morphisms we choose to care about are algebraic; that is, that they are ultimately built out of polynomials. This is what distinguishes it from, say, differential geometry, where the morphisms we choose to care about are built out of smooth functions instead.