Can anyone explain how this battery charger works?

Maybe it's an AC battery? They are very useful for grid storage applications as long as you can control the phase.

What makes you think that component is a capacitor? I'm assuming you drew the schematic.

It's also not exactly what I''d call a charger. It's more of a charging indicator circuit. It takes DC input and puts it straight to the battery through a diode. The rest is there apparently for the two LEDS that show power on and charging.

It is called "resonant charging" and the current is pulses - so it DOES flow through the capacitors

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A transformerless power supply (TPS) is basically just a voltage divider that takes the 115 or 220 VAC from your wall and divides it down to whatever voltage you want. If that voltage needs to be DC, it is rectified through a few diodes, and maybe regulated to a maximum voltage but we?ll get to that in a minute.

Normally, DC voltage dividers are made with a pair of resistors. Combined, they define the current flowing through the path, and the top resistor can then be chosen to drop the difference between the input voltage and the desired output. If, in our case, that difference is some one or two hundred volts, even if it only has to pass a few tens of milliamps, that resistor is going to get hot fast.

A better component to use in the top of the divider is a capacitor, with its reactance chosen to give the desired ?resistance? at whatever the mains frequency is where you live. For example, say you want 25 milliamps out at 5 V, and you?re in America and need to drop 110 V. R = V / I = 4,400 O. Using the reactance of a capacitor, that?s C = 1 / (2 * pi * 60 Hz * 4400) = 0.6 µF. If you need more current, use a larger capacitor, and vice-versa. It?s that easy!

A fully elaborated TPS design requires a few more parts. For safety, and to limit inrush current, a fuse and a one-watt current-limiting resistor on the input are a good idea. A large-value discharge resistor in parallel with the reactive capacitor will keep it from holding its high voltage and shocking you when the circuit is unplugged.

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Did you even look at the circuit? There is no pulsing, it's DC input, DC output. And Commander acknowledged that what he drew as a cap, isn't a cap at all.