Why do airplanes circle three times in a holding pattern? I tried to look for information about it, but I did not figure out why a pilot would specifically do it three times?
They don't, necessarily. A holding pattern is a way for air traffic control to slow down an aircraft when an airport does not have enough capacity to meet the number of inbound aircraft (essentially, a traffic congestion). It can also be used in other situations, for example when the weather is too bad for an aircraft to land, and they need to wait for improvement before making an approach.
A standard holding pattern takes 4 minutes to fly. One minute inbound- and outbound legs, and one minutes in each end to turn 180 degrees (a standard turn is 3 degrees per second).
So, if an aircraft needs to be delayed for 12 minutes, 3 turns in a holding is appropriate. However, the actual required delay depends entirely on the situation, and so there is no standard number of turns in a holding pattern - it all depends on what is required.