Why does string theory have such a huge landscape?

Who wrote that passage? It contains some misunderstandings.

All I know is that $10^{500}$ is a very large number.

It is a finite number. How many theories do you know which have a finite number of solutions? Have you tried to count the number of solutions of plain Einstein-Yang-Mills-Dirac-Higgs theory without its string-theoretic UV completion? There are not only infinitely-many solutions, there is a hugely infinite-dimensional space of solutions. This is the usual state of affairs for most every theory of physics ever considered. String theory is special in that it puts many more constraints on the solutions, such as to even leave just a finite number (under some assumptions).

What exactly is a 'solution' in string theory?

A background for perturbative string theory is a choice of 2-dimensional superconformal QFT of central charge -15. This can be interpreted as describing an effective target space geometry which is a solution to a higher dimensional supergravity theory with higher curvature corrections. A "solution" to string theory is a solution of the equations of motion of that. At least without non-perturbative effects taken into account.

See on the nLab: landscape of string theory vacua for more.

I thought string theory was supposed to be finite, why do perturbative series still diverge?

String theory is thought to be loop-wise finite, thus being a renormalized perturbative theory. No sensible remormalized perturbative QFT can have converging perturbation series. The perturbation series must be an asymptotic series to be realistic, and it comes out exactly like this in string perturbation theory.

See at the String Theory FAQ on the nLab the item Isn’t it fatal that the string perturbation series does not converge?

Is there any experimental technique to limit the number of 'solutions'? Will experimental techniques be able to pinpoint a solution within present day string theorists' lifetimes too?

Models that have been and are being built in string phenomenology approximate the standard model to more detail than probably most people are aware the standard model even has. Check out some of the references there. Given the slow but continuous flow of new articles on these matter, one sees that some people are slowly but surely working on improving ever further. Check out the references at string phenomenology.

[edit: I have now added a corresponding item to the nLab String Theory FAQ: What does it mean to say that string theory has a “landscape of solutions”?]

One way to understand the landscape is as the space that arises from compactifying a higher-dimensional theory in multiple possible ways. In 10-dimensional string theories, for instance, you usually compactify your theory on Calabi-Yau manifolds with 3 complex (6 real) dimensions, which is convenient because it gives you a compactified theory with $\mathcal{N}=1$ supersymmetry (you'd choose a different manifold if the world happens to have a higher level of supersymmetry). Analogously, in 11-dimensional M-theory, you compactify on 7-dimensional manifolds with $G(2)$ holonomy, which is just as convenient.

But there are a large number of such manifolds, so you have a large number of 4-dimensional solutions.

The way you test for the right vacuum is by looking at the phenomenological predictions made by each theory, or classes of theories -- mass ratios and such -- and check if they meet our observations.