How do you check that the open sourced code is the same one that you are installing from the extension repository and actually running?
Same argument can be applied to all closed source software.
In the end its about who you trust and who needs to be verified and that is relative, subjective, and contextual... always.
So unless you can read the source code and compile yourself on a system you built on an OS you also built from source on a machine built before server management backdoors were built into every server... you are putting your trust somewhere and you cannot really validate it beyond wider public percetptions.
Extensions are local files on disk. After installing it, you can audit it locally.
I don't know about all operating systems but on Linux they are stored as .xpi files which are zip files. You can unzip it.
On my machine they are installed to $HOME/.mozilla/firefox/52xz2p7e.default-release/extensions but I think that string in the middle could be different for everyone.
Diffing it vs what's released in its open source repo would be a quick way to see if anything has been adjusted.
I'm currently in the process of setting that up for the one I'm building, because this transparency is very important to me) and it is a pain in the butt to do so. You have to go through a few verification processes at Google to get the keys approved.
There’s always a possibility of problems along the chain. You are reducing your risk not eliminating it.
Got to say, mischaracterising a neutral question as a nihilistic comment doesn't do anything for me.