Moonrise delay
Why does moonrise get later each night?
The Moon moves east along its orbit while Earth turns. Earth has to spin extra — about 50 minutes — to bring the Moon back to the same place on your sky. That extra is why moonrise (and moonset) slip later day by day.
The same 50 minutes as the month
A synodic month is about 29.5 days. Spread that extra lap around the sky over 24 hours and you get roughly 50 minutes. The delay is not a separate mystery. It is the month, counted per night.