If you've ever hung wallpaper, you know it can get pretty intense when you've spread the wallpaper paste and you then have to hoist it up the ladder to the corner of the ceiling to hang it on the wall. It is tricky to get it perfectly lined up with the vertical plumb line, and get it glued evenly to the wall, without bubbles or lumps. And it all has to be done in fairly short time, before the paste dries. It's a tricky job at the best of times, when you have two arms. Imagine the hassle of doing all that at once -- with one arm!
'A new brush sweeps clean' means that someone with a new perspective can make great changes. However, the full version is 'a new brush sweeps clean, but an old brush knows the corners', which warns that experience is also a valuable thing. Sometimes 'broom' is used instead of 'brush'.
This means that something is worthless and dates back to when someone would travel around the countryside repairing things such as a kitchen pot with a hole in it. He was called a 'tinker'. His dam was used to stop the flow of soldering material being used to close the hole. Of course his 'trade' is passť, thus his dam is worth nothing.