Yuk - what a horrible circuit!
The answer to your question is that C1 is there to bypass R1 and couple the
output of the microphone onto the base of T1.
Basically, circuit operation is as follows:
T1 and its associated components provide voltage gain to the signal from the
microphone and T2, together with C4 effectively provide half-wave
rectification of the amplified signal on T1 collector.
The larger the output from the microphone, the more positive the voltage on
T2 emitter will be and hence the more LEDs will be illuminated, starting
with LD1, turned on by T3, and progressively lighting each LED to the right
until all are lit.
Each of transistors T4 thru T7 has a threshold voltage one diode drop above
the one to its left.
I assume from your question that you are a relative newcomer to electronics
and, if so, I would not recommend this circuit as an indication of how to do
It is very poorly defined in terms of its DC bias conditions and both its
zero level and sensitivity will be prone to temperature drift. Furthermore,
RV1 which is there presumably to set the zero threshold level changes the
sensitivity at the same time - awful!
Hope this helps