The 8259A Programmable Interrupt Controller (PIC) allows for more interrupts than the 8085 processor's 5 hardware interrupts. It can handle multiple interrupts simultaneously with different priorities. It manages 8 interrupt lines and their priorities through control registers. It interfaces with the CPU through control words that initialize it and perform operations like masking interrupts. Cascading multiple 8259As allows for up to 64 interrupt levels to be managed.