What you have there are punch-down connections, where the wire is forced into a slot that cuts through the insulation and makes the electrical connection.
To replace them with your new screw-terminal type jack:
- Pry the wire out of the slot.
- I would suggest that you cut off the part of the wire where you can see the insulation is nicked, because the metal will also be nicked and might be fragile there. This might not be necessary.
- Use a wire stripper to expose some bare wire.
- Hook the wire around the screw and tighten it down.
To keep the jack wired the same as before, make sure you pair your blue-with-white-stripes wire with the jack's red wire and the white-with-blue-stripes wire with the jack's green wire.
However, there is something weird here: the blue-with-white-stripes wire seems to be looped through the jack and on to a splice with the green wire. This is strange, because wires should always be paired and the white-with-blue-stripes wire isn't doing the same thing. But perhaps the other splice is joining 3 wires (I can't quite tell). In any case, make sure to preserve any wire joins, as they might be part of the wiring for another jack on this same phone line. You may wish to replace the loop with a new 3-way splice, or you can try cutting the loop and putting both ends of it under the red screw.
The exact way you make the connection doesn't matter too much, as long as everything has continuity that did before.