Set width and height to 100 % and if you want to maintain a 16:9 aspect for instance you also need to add a top-padding of 56.25 %.
You can read more behind the math of maintaining aspects here:
Currently the built in background video component in Webflow don’t seem to be working
on mobile devices though. I’m waiting for someone from the Webflow team to reply in this thread that I created: Background video don't play on mobile devices 2020 - #6 by Olle
It should be working since a 2017 fix where Webflow added two html attributes to the component but don’t seem to be working currently, at least for me.