Imagine the following line as a part of the Moon's orbital path across the sky from west to east. The Moon travels from West to East across the sky, and makes one full journey with respect to the stars every 27.3 days ( a Sidereal Month):
Now, if it takes 27.3 days to travel once around the Earth, it must travel 360 degrees/27.3 days = 13.18 degrees/day to the East. This means on the next night, the Moon is located 13.18 degrees to the East from last nights location:
This means that the Earth has to turn an extra 13.18 degrees so that so that tonight's moon is in the same sky posiyion as last nights moon. If last night the Moon was just at the eastern horizon, tonight at the same time it is 13.18 degrees Below the eastern horizon.
Now, how long does it take the good ol' Earth to turn 13.18 degrees? Well, in 24 hours, it turns 360 degrees, so in (13.18/360)x 24 = 0.88 hours or 52.7 minutes, the sky rotates the extra 13.18 degrees.
Now, why do we use the time of the Sidereal month instead of the Synodic month which is 29.53 days? Because we are interested in the Moon's relation to its position relative to the background sky ( Sidereal) not whether it is in the same orientation with respect to the Sun and Earth. A Synodic month separates one New Moon from the next New Moon, or any corresponding similar lunar phases on any two cycles. If we were to use the Synodic month, we would get a lunar shift of 12.1 degrees per day, and that the Moon would rise 48.7 minutes later each night. The average of the two is 50.7 minutes. The difference between the two is 4 minutes, which is just the amount that the Sun has moved to the East in ITS motion along the sky. This emphasizes that Sidereal time does not depend on the location of the Sun, but Synodic does!
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