Electronic – How does this current limiter work? – and why doesn’t it

The transmitter does indeed turn the DC into an oscillating signal, however it's worth pointing out that it's more of a sinusoidal signal than a typical square wave. That's the purpose of the capacitors- to resonant in a "tank" circuit with the inductors. That allows more energy to be transferred and there is less energy in higher harmonics (if the oscillator runs at 50kHz, a square wave would have a lot of energy at 150kHz and higher frequencies which could cause interference).

The 1N4001 is not rated high enough voltage (it's only a 50V part) compared to what the designer says is required (400 PIV), also it's extremely slow in reverse recovery. In fact, most devices made these days that are marked 1N4001 are actually identical to 1N4004 or (400 PIV). 1N4005 and higher use a different chip internally. The trr (reverse recovery time) of the 1N400x series is not specified, but it's actually in the microseconds compared to 75ns for the UF4007 and 50ns for the UF4004. That would cause a lot of trouble if the waveforms were more square, but at 50kHz sine wave you may be able to get away with it with some loss of efficiency (even though officially it meets neither of the requirements that are stated(!)- it's 50V rated not 400PIV and it's anything but fast- maybe 100x slower than he asks for).

Ideally, both diodes and the four diodes in the bridge rectifier should be UF4004s.

You have to read the MBR120 datasheet carefully, that Vforward of 0.4 V is at 1 A. This circuit is not going to draw 1 A.

My guess is that it draws less than 100 mA, now let's look at the datasheet again.

At 25 Celcius Vforward is less than 0.2 V. Only under very cold conditions will it increase (but in the cold the battery will give up also so this is a moot point).

So this means that the minimum required Battery voltage will be around 3.3 + 0.25 + 0.2 = 3.75 V, a lot better than your 4.0 V.

Even if the battery is slightly below 3.75 V, it does not mean the circuit will stop working immediately. The regulated 3.3 V might drop a bit and that is no problem either as the Bluetooth module can work down to 1.8 V and the Atmega32 can work down to 2.7 V.

My guess is that the module will only stop working if the battery voltage drops below 3.3 V or so. And then the battery is empty.

Also, have a look at a discharge curve of a LiPo battery:

We discharge the battery slowly so we can use the black 0.2C curve. Note how you would have used 90% of the capacity if you would discharge to 3.6 V (which is a safe value).

So it all looks reasonable to me !