How do incompressible strings and random strings share the same properties?
The theorem says, in effect, that incompressible strings are a deterministic model of random strings. If you have some property that you can prove using random strings, you can also prove it using incompressible strings instead. If you have some construction involving a random string, you can make it somewhat more explicit using an incompressible string.

Another deterministic model of randomness, which is very useful in theoretical computer science, is expanders. These are models of random graphs with respect to a lot of properties. In many cases, instead of using a random graph you can use an expander. Since we can construct expanders explicitly, this allows us to derandomize many randomized constructions.

While the same cannot be said for incompressible strings, they are useful in some circumstances. Their use in lower bounds is known as the incompressibility method, which I encourage you to look up.