One awesome thing about WebAssembly is that it makes a lot of sense as an extension language. Instead of building around the JVM or .NET to get access to languages on those platforms and instead of standardizing on one embedded language (Lua, Python or JavaScript), I can use anything that supports WebAssembly and double dip with that same ecosystem working in the browser. And that list of supported languages will only continue to grow since we finally have an alternative to compiling to JavaScript.
Now, I don't think it'll "take over" the way C has, but I also wouldn't be surprised to see it edging out other VMs in many contexts, like the Lua or Python VMs, which may end up porting to WebAssembly instead of trying to justify another dependency for your app.
I hope we'll finally get to the point where I can choose what language to use for a specific task based on the merits of the language instead of making the larger argument that it's worth getting it to work in our existing app.