In practical terms the language of the browser is JavaScript, but what if JavaScript doesn't do what you need, or it isn't fast enough, or you simply don't like JavaScript. Instead you want to write some application in Rust (just using that as an example). The language of the web is still JavaScript and not Rust so your new application will not run directly in a web page. That's unfortunate, because the web is a fantastic distribution platform because things that run in web pages do not require a separate installation step or a separate download step, since web page assets are downloaded automatically as called by the web page. If you compile your Rust application to WASM it can run in a web browser.
To be clear though WASM isn't replacing JavaScript or even in any way competing with JavaScript, because that is certainly what it sounds like. WASM is a separate and isolated execution instance in the browser, even from the page that contains it. JavaScript, on the other hand, executes directly in the context of the page that requests it.
You might hear some discussion of wanting WASM to replace JavaScript, but that is clearly stated by the WASM project as not a design goal. You might also hear that if only the containing page's DOM were available to WASM it would replace JavaScript, but that isn't a design goal either. That part about the DOM is the confluence of a misunderstanding about the DOM and about WASM's security goals of a separate execution context. There is work, however, on providing some Web API capabilities to WASM so that a WASM instance may have some awareness of its execution context or to allow some manner of external interaction of the WASM instance from the surrounding page. I don't know the status of that work though.
* WASM project site - https://webassembly.org/
* Web APIs, a list of de facto standards for interacting with web browsers from JavaScript - https://developer.mozilla.org/en-US/docs/Web/API
* DOM specification, the DOM is a language agnostic dynamic tree model that represents markup and containing content - https://dom.spec.whatwg.org/