How does this PIC board achieve RS-232 communication without a line driver?
The logic-level signals to/from a UART are inverted relative to RS-232 levels.  The transistors do the inversions.

Diode D3 takes power from the normally-negative RS-232 RX line to charge C9 which then provides the negative voltage for the RS-232 TX line.

R18 and R15 limit the base current to the transistors, and R27 and R16 ensure that the transistors do turn off, by pulling the base towards the emitter.