> Who is F-Droid? Why should I trust them?
For the same reason you trust many things. They have a long track record of doing the right thing. As gaining reputation for doing the wrong thing would more or less destroy them, it's a fair incentive to continue doing the right thing. It's a much better incentive that many random developers of small apps in Google's play store have.
However, that's not the only reason to trust them. They also follow a set of processes, starting with a long list of criteria saying what app's they will accept https://f-droid.org/docs/Inclusion_Policy/ That doesn't mean malware won't slip past them on occasion, but if you look at the amount of malware that slips past F-Droid and projects with similar policies like Debian and compare them to other app stores like Google's, Apple and Microsoft there is no comparison. Some malware slips past Debian's defences once every few years. I would not be surprised if new malware isn't uploaded to Google app store every few minutes. The others aren't much better.
The net outcome of all that is the open source distribution platforms like F-Droid and Debian, that have procedures in place like tight acceptance policies and reproducible builds are by a huge margin the most reliable and trustworthy on the planet right now. That isn't saying they are perfect, but rather if Google's goal is to keep their users safe they should be doing everything in their power to protect and promote F-Droid.
> How do I know they aren’t infiltrated by TLAs? (Three Letter Agencies), or outright bad-actors.
You don't know for sure, but F-Droid policies make it possible to detect if the TLA did something nefarious. The combination of reproducible builds, open source and open source's tendency to use source code management systems that provide to audit trail showing who changed every line shine a lot of sunlight into the area. Sunlight those TLA's your so paranoid about hate.
This is the one thing that puzzles me about F-Droid opposition in particular. Google is taking a small step here towards increasing accountability of app developers. But a single person signing an app is in reality a very small step. There are likely tens if not hundreds of libraries underpinning it, developed by thousands of people. That single developer can't monitor them all, and consequently libraries with malware inserted from upstream repositories like NPM or PyPi regularly slips through. Transparency the open source movement mostly enforces is far greater. You can't even modify the amount of whitespace in a line without it being picked up by some version control system that records who did it, why they did it, and when. So F-Droid is complaining about a small increase in enforced transparency from Google, when they demand far, far more from their contributors.
I get that Google's change probably creates some paper-cuts for F-Droid, but I doubt it's something that can't be worked around if both sides collaborate. This blog post sounds like Google is moving in that direction. Hear, hear!