When engaging the brake lever: the cable loosens and doesn't stay in the other side (where the twisty bit is)
It looks to me like you are pulling too much cable. It even looks like your brake lever is close to being able to bottom out against your levers. In normal operation, brake levers do not come out nearly as much as you have as in the picture. 

First thing to check is to make sure your brake pads are not worn out. If they are, replace them. Then re-adjust your brakes so that you have as little clearance as possible without any rim rubbing. If your wheels are so out of true that doing so is very difficult, it would be worth resolving that.

Then, check your brake arm tensioning spring. This will help pull on the cable and to keep the cylindrical cable end seated in the slot on the brake lever. See if you can move the spring to a higher tension setting. Your frames brake post should have three holes in it, you can adjust the tension by removing the brake arm and replacing it with the spring in a different hole, and your brake arm may also have a tension adjustment screw to fine tune it. 

And just as a sanity check, make sure the cable isn't slipping through the cable binding bolt when you hit the brakes.