I'm not 100% sure about any of this (I've only listened to most of the video above), but I think the gist is that WebAssembly provides a compile target for applications, meaning you can pre-compile an application so that your source is downloaded essentially as machine code. Furthermore, that means your compilation can perform as many optimizations that you care for, because it's happening ahead of time (not on the user's machine).

Without WebAssembly, browser vendors have to strike a balance between optimizing the JS as much as possible (for increased performance), and running the script as soon as possible (so the user isn't waiting too long for execution to begin).

edit: Demo of this in-browser video editor charts a FPS difference between the JS and WebAssembly implementations. https://github.com/shamadee/web-dsp

Compressed download size is indeed only slightly smaller (mostly around 10%), but the parsing is much faster, even noticeable for fairly small demos.

(edit: replied to the wrong comment, apologies)