There are metal contacts that you push the solid conductor into that cut into the insulation and make electrical contact (insulation displacement).
Of course, you need to know the correct order/color code of how to connect them...hopefully you can derive that from the damaged connector you're looking at now.
As far as finding a new pigtail for the camera goes, even if you find one, getting the old pigtail out of the camera case without destroying anything is pretty difficult. The punchdown jack is probably the most practical way to save the camera. There's also the possibility of cleaning up the connector in the picture. Worth the try if it's not damaged too seriously.
The camera connectors with their hoods been doing well for me at being rain-proof, but you do have to pay attention to details. I imagine general climate (e.g. desert vs. swamp) has a lot to do with it too.
1. Don't forget the gasket between the halves.
2. Some of the supplied gaskets have square shoulders. Make sure they are not twisted around the barrel.
3. Make sure the halves are mated firmly.
4. Make sure the gasket around the incoming network cable is firmly compressed.
5. Align the connector straight up and down, with the camera end up.
Against the forum's wisdom I've found no need to use dielectric grease on the contacts with these connectors. I've experimented with using grease on the gasket surfaces and coax seal where the barrel halves are mated, without knowing if that was necessary and/or helpful. Underground connectors have been a whole different story and need a lot more attention. I went through a few years of frustrating learning curve with them.