I would like to plug in my outside nest camera to the outside GFCI circuit. It is a weatherproof outlet but I can't get the flaps/holes on the bottom or side to open.
How to do that?
Here is a picture:
Typically a utility knife, a pair of pliers, or both. Cut, then twist, if using both.
The plastic is made thin at the edges to be broken out, but thick enough that the port remains intact until someone decides they actually want it open. They don't slide or otherwise open in a reversible way - they are literally broken out, and the hole will be there forever after.
Stick to using the ones on the bottom as it is mounted, for the least impact on weatherproofing.
If you want this to stay waterproof-ish and the cabling is thin enough, I'd suggest to drill a hole big enough to accommodate a cable gland big enough for the cable and just use cable glands (the plastic screw-on-with-O-ring types) instead of knocking out. Worked like a charm for me. Sometimes you can even get glands that simply match the size of the hole you'd make by doing full knockout instead (i.e. have the same radius), and you can just drill a hole that matches the knockout radius itself and leave the rest of the plastic be.
Alternatively - drill a hole in the knockout, insert the cable and use silicon/epoxy to cover the rest of the hole. If there is no mechanical stress/tension on the cable, and if the hole is placed at the bottom of the box, it will be waterproof enough for most cases.
I don't exactly know the name of this, but thanks to a precocious 2-year-old, I noticed that some of these weatherproof receptacles come with a sort of rubber brush cable pass.
You can barely see it on the left side of this picture (and I believe the same cable pass is on the bottom). When you remove the hard plastic you can feed a cable through the rubber brush without completely destroying the weather-proofing (probably best to still only expose the bottom).
It's unclear in your picture if there is a similar rebate for a cable-pass/grommet/gasket but if there is, you could try to find a replacement that fits it for a long-term solution after if/when the plug is removed.