The circuit provided an voltage that oscillates between about +1V and +11V at the common point between the two output transistors. The voltage is able to source and sink current.
When it is high, C3 is charged with the transistor end positive, while D1 takes the charging current and keeps the diode end of C3 down to about 0.7V, leaving around 10V of charge on the capacitor.
When the output of the oscillator is low, there is still 10V of charge on the capacitor. It now has its transistor end at about 1V, so its diode end is down at about -9V. The diode drop of D2 means C4 gets charged to about -9V. It will take a few cycles to get there as C4 is charged from C3.
This is a standard circuit. It is like what is used in a MAX232. If you have any oscillating voltage on the positive side you can omit the oscillator and just use C3, C4, D1 and D2, as long as the oscillator can provide the current that you need.