Well it can't be received with an UART easily. One thing you could do with Arduino is to feed the data to Timer Capture Input pin and triggering on the edges and every time an edge happens make it trigger on the other edge and look at timer value how much time was between edges to decode the bits.
I don't know who designed that board, but it has several questionable connections.
First, CH340C chip datasheet requires V3 pin connected to VCC if VCC is 3.3V. On the board it is permanently connected to decoupling capacitor, which corresponds to 5V configuration.
On the other hand, the LEDs are connected to 3.3V supply, so they will (maybe) operate in ...
In addition to what Justme recommended: at 62.5 kbd, you can just probe the input pin at say 145 kSamples/s and do all the de-manchestering by converting the samples to +/- 1 and then multiplying consecutively.
Timing offset, rate error and sync loss nonwithstanding, of course.
On faster processors, just sufficiently oversample the pin (say, at 500 kS/s), ...
After a chat with OP to debug connections, here is how the problem was (somewhat) solved:
HC-05 should be connected:
VCC and GND to the Arduino's VCC and Ground respectively
TX of the Arduino (the designated TX pin; the second argument of the SoftwareSerial declaration) goes into a resistive divider of 1k:2k, and the output goes into the HC-05's RX pin