What is the function of these extra capacitors around the bridge rectifier?
Those extra small value capacitors are to avoid the rectifier generating RF interference.

When a diode has been conducting and then current reverses it can stop conducting very suddenly when the charge carriers are depleted.  This can occur extremely rapidly and generate interference to many MHz.

These sudden changes in current can cause interference to other parts of the circuit or even other equipment. It is modulated at 60Hz or 120Hz (US power frequency) so would be heard as a buzz in audio equipment. 

The usual avoidance technique is as shown in the circuit. Ferrite beads may also be required in stubborn cases.

The technique can be used intentionally to generate high frequencies or narrow pulses, especially when high frequency devices are not easily available.

Step Recovery Diode