The circuit consists of the launch timer itself and an automatic-off timer. When power is applied to that IC, the countdown LED's sequence is on until they are all lit. When the last one LED1, is fully lit, transistor Ql saturates, energizing RY2. When that happens, a circuit between the lantern battery at the launch pad and the nickel-chromium wire is completed; the wire heats up as before, and the rocket is launched. Resistor R4 and capacitor C3 determine the countdown timing; with the values shown it should be approximately 10 seconds. Resistors R3 and R5 set the LED brightness. Safety is of the utmost importance.
That's the purpose of the second half of the circuit. When RY2 opens, the current flow to Q2 is disrupted. But, because of the presence of R2 and C4, the transistor remains saturated for about 3 seconds. After that, however, the transistor stops conducting and RY1 is de-energized. That cuts off the power to the rest of the circuit, and RY2 de-energizes again, breaking the circuit to the launch pad. Switch S3 is used to reset the countdown. Once that is done, pressing Sl starts the launch sequence; the rest is automatic. Switch S4 is used to latch RY1 manually if needed.