Explain this circuit
That's a super-regenerative receiver, very commonly used in toy car receivers.
That construction technique may look messy, but it is OK for 27 MHz frequency. The major difficulty is ensuring:


the 6 turn coil on the ferrite slug resonates (with 47pf + a bit more) @ 27 MHz.
the whole thing oscillates at a frequency usually in the 20kHz - 50 kHz ballpark.


That 70-turn coil attached to the emitter is involved with the 50 kHz oscillation. Its value might be in the 3uH to 5 uH ballpark. You can use a molded fixed inductor here.
 The 47 pf, 39pf, plus antenna length affect the 27 MHz. resonant frequency. Most often, the collector coil has an adjustable slug to set this frequency very close to 27 MHz. These receivers use a short antenna that appears as a capacitive reactance - its length affects 27 MHz frequency. Your receiver is likely insensitive, because this frequency is somewhere else.
An oscilloscope is very useful to ensure that this oscillator is oscillating at both high frequency (27MHz) and low frequency (50 kHz) at the same time. 
You should notice that the 50 kHz frequency becomes higher when a 27 MHz signal from a transmitter is present. Look at the transistor emitter to see this.