Web development is horrifically, unbelievably bent around two facts:
1. There has been no convenient, highly popular and supported language for development that also has best-of-breed performance. You can have one or the other. (Yes, I've probably heard of your obscure fast-as-C language that has a great web stack... to a first approximation, nobody else has.)
2. Web development languages have been profoundly, deeply single-threaded, to the point that the web world began to take it as an unmitigated virtue that a page started with a request, generated a response, and dropped everything in the world that it knew at the end of the request. State is something that must be managed carefully, but especially in this era of people writing web applications that may literally have gigabytes available per concurrent user (obviously, not everybody, but this is a non-trivial use case!), the statelessness dogma has seriously broken people's mental models vs. what the hardware is actually capable of.
With Rust, you should be able to write fast, concurrent web applications that actually do something concurrently and aren't constantly marshaling across requests for the client to re-initiate AJAX calls or other such silliness, it'll be fast, and with its emphasis on memory safety, statically safe as well (something that you're not going to get very easily out of Java, and certainly not from a programmer just sitting down and pounding out some Java). It isn't going to be the solution to every problem, but it's going to be the solution to more problems than people realize right now, because we've so deeply internalized our essentially-PHP models of how the web works that we can't hardly even see how things ought to actually be working.
Rust isn't the only language biting at this apple. Go is obviously another one. But Rust has a pretty decent shot at taking a big chunk, if it plays its cards right.
You say the popular web development platforms are all dynamically typed... I'm increasingly coming around to the position that was an accident of history, rather than an immutable fact.