What is this plastic tubing and why would water be dripping from it?
It looks to be a saddle valve which is used often because it means that you can add a tap for a fridge's ice maker to an existing water line without cutting into them. However they can start to leak and they don't get a perfect seal into the pipe when turned off either.
If you find it leaking you should be able to replace it with a proper tee and valve to fix the leak.