Oh dear. That was botched by a hobby installer. "Internet coming out" can’t work here because it first has to somehow get into your home network, and then because of the socket wiring. These are ISDN sockets, which you can also see from the terminal strip that only has 4/5 and 3/6, so 1/2 and 7/8 are missing. For Ethernet, you would need 1/2 and 3/6 (or all four pairs for Gigabit Ethernet). Forget the nonsense about shielding making it unusable; I work professionally in the industry and have seen more perfectly functioning internet over doorbell wires in practice than even the well-informed ITK reader can believe.
You can still use the socket insert if you want to use it for analog phones, which are by no means obsolete yet—unlike ISDN. The cables you see are two different ones, and someone must have had a reason for that, because actually only the "smaller" one of the two would have been necessary here. The questions of where they come from (and by which route(s)) would be worth investigating.
The cables are, on the one hand, classic telephone cable, of which only the first pair was used (red/black), and a 6DA cable, also TK (or for moderate demands also IT) cable. You can keep the first as a reserve behind the socket; you can use the larger cable (for Gigabit Ethernet only for one socket). The pairs belong together as follows: red/blue, white/yellow, white/green, white/brown, white/black, and white/blue, cf. e.g. tocker (de). Together these are six twisted pairs, of which you only need four for one socket. Mixing with the other would also be possible, but that wouldn’t be my first choice.