How do I replace the feet?

You should replace missing or worn feet right away.  The metal on the washer against the metal of the sink can cause a problem with the sink if ignored.

Look at the bottom of the foot to determine if you need a Philip or straight head screw driver.  Loosen the old foot and simply screw the new foot on.  Seat foot against plate tightly.  4 feet per machine.

Cause or how do I avoid this problem from repeating?

As in any appliance there is some vibration.  While lost feet are generally not a problem you can place a dab of silicone calk on the threads and this will effectively glue the foot on while still being able to remove it in the future.