If you don’t need the splitter that comes out of the wall valve, first of all I’d get rid of it and connect the washing machine tube directly to the valve on the wall (looks like your not using the one on the right in the first photo) then you should be able to just open the main valve and get water into the washing machine, if one side of the tube leaks, try tightening it, if it still leaks unscrew it from both sides and check the rubber washer that’s inside the tube head, if it looks worn out you can try adding some teflon tape (if you have any) otherwise just bring the tubing to a hardware store, buy a new one and install.