I have 220v coming into my garage, obviously for the ancient welder outlet, utilizing the old 50-60's style fuse box. There is also 110v coming off the box, but I am not sure how it gets down to the 110v. I am installing a new breaker box, how do I get from 220v to 110v for my outlets and lights?

You are getting 220V by using 2 120V lines and a neutral. To only use 120V, all they are doing is using power from one of the two hot legs and the neutral.