What are the purposes of these capacitors?
Capacitor C1 seems strange, also I don't see any resistor for the LEDs which makes me think the C1 is actually a resistor (or it should be).

Explanation: See comments of ktilcu/KingDuken below: It looks like a PTC, (in this case) stands for "positive thermal coefficient" which typically is an algorithm used for measuring heat. The higher the heat, the more resistance you have. "C1" could actually be a thermistor that could create a current limit.

Capacitor C3 is a so called bypass capacitor going from VCC to GND and is typically 100 nF. It keeps the voltage more stable in the circuit and is typically used for ICs. Normally it's placed as close as possible to the VCC (+) side.