Washing machine pressure switches directly control the hot and cold water inlet solenoids and so are rated for quite high currents - 15A at 230V AC is typical. So for low-voltage DC applications, the switches can certainly cope with (say) 5A. This means that, for most loads, a relay won't be needed.
The on-off cycle in the master timer should go on only for the set time (here it is 18 minutes). Once the master timer goes off, the cycle should stop. To achieve this, the outputs of both the timers are connected to NAND gate N1 (IC3), which gives a low output only when both the timers are giving high outputs.The output pin 3 of N1 is connected to relay RL1 via pnp transistor T1, so the relay energises only when the output from NAND gate N1 is low. As the mains 220V line is taken through relay RL1,the monitor turns off during the 3-second off period after the set time of 10 seconds is over.