This is a solar energy charger and have a PNP 2N5401 transistor

can anyone explain to me about how this circuit can work ? and the LED can light up when battery is implement.

 

The "top" circuit has something wrong ....... What is source 1? Source 2? Single line sources?
Re trace the circuit.

The "bottom" part seems OK.

No 1.2 V NiMH battery1 , no Q1 base drive, no led1 drive.
NiMH battery1 in position, yes Q1 base drive (battery charge current partly thru' R2, partly thru' Q1 base), led1 driven to light.

and so for q2 q3 q4.

 

It wont work, the nimh batts wont charge.

 

Yes, at 3 mA!

The OP wanted to know how an inserted battery lights the LED

 

Yes, at 3 mA!

How did you get 3 mA, I got less than 0.06mA.

Are D1 to D4 diodes or zener diodes? If zener, what's the breakdown voltage? If they are regular diodes, why are they there?