Think of the clock as a circle where 0 degrees resides at 3 o'clock. If you move the hour hand to 12 o'clock and leave the minute hand on 3 o'clock, you've created a 90 degrees angle between the hands (because the hands make a right angle and a right angle is 90 degrees).
Now if you split the clock into 4 90 degree quadrants and use counter clockwise as positive angle rotation (standard math convention), you can see that moving the minute hand from 3 o'clock counter clockwise all the way around to 6 o'clock, passes the minute hand through the first right upper quadrant, the second left upper quadrant and the 3rd lower left quadrant if you were facing the clock head on.
These 3 quadrants is 90 + 90 + 90 = 270.
The hour hand, starting from 3 o'clock is moved to 2 o'clock which is still on the upper right quadrant. Since each quadrant is 90 degrees, you must divide 90 degrees by the numbers in the quadrants. Since their are 3 numbers, 12 to 1, 1 to 2, 2 to 3, 90 / 3 = 30 for each movement of the hour hand to the numbers.
Since you move the hour hand up one, you're subtracting 30 degrees from 270 since leaving the hour hand at 3 would make the angle between the hour and minute hand 270.
So, 270 - 30 = 240 and that's the answer.