AC to DC without transformer. How does this thing work?
A capacitor is an impedance and at (say) 50 Hz, a 1 uF capacitor will have an impedance of 3183 ohms - if you put this capacitor directly across the AC (at 220 V), the current that would flow is about 70 mA.

OK, so it's not directly across the AC supply because it feeds a bridge rectifier and battery but it will provide a decent enough source of current to charge the battery. The voltage is limited by the battery itself.

Be aware that this type of circuit is not isolated from the incoming AC and stripping it down is OK providing you don't switch it on while still stripped down&mdash;the voltages are lethal.

Here's something like the circuit it uses: