It uses the MCP73861 or MCP73863 Li-ion battery charger chip manufactured by Microchip.
Step 1: The USB power connector
Step 2: The circuit board
The MCP73861 or MCP73863 (they are similiar, with only slight differences which do not affect the circuit connections) is available in a small leadless package.
The difficulty? There are no leads to solder to. The advantage? There are no leads to break!
The ic is placed so that its connection side (the side with solder pads) lines up with the copper side of the board and it is then fixed in position with epoxy or some such glue.
Step 3: Soldering the integrated circuit
I find it helpful to flatten the wire with pliers before soldering, so that it stays in position without any tendency to roll about. Some of the leads go to the same node and these are conveniently placed together.
After the leads have all been soldered, the space in between the leads are cut out to form islands and the other components are soldered to these islands of copper.