Hello, thanks for posting, and thanks for including the good photos. The LEDs are arranged in a scheme called Charlieplexing, which makes it possible to drive many LEDs using few pins, but the downside is that any short circuits make the LEDs light up pretty randomly. From your photos (thanks again!) it looks like there might be a couple short circuits between LED pins.
You can remove some of the excess solder by re-heating each pad until some solder attaches to the iron, then wipe off the solder onto a wet paper towel or napkin. Ideally you want to have no short circuits between neighboring solder joints. Keep doing that to remove all the extra solder from each connection, and then trim down any excess wire leads exposed. You can also use your iron to heat up the point where each wire lead passes through the board, so that you are heating up both the wire and the PCB pad around the wire, that will help draw the solder into the joint and stop pooling in a ball on the board's surface.