That is a very simple geometric problem. Imagine a globe on your desk and two table tennis balls. You hold one ball at one side of the globe and the other ball on exactly the opposite side of the globe. You notice that these balls can't 'see' each other, no matter how large or small the distance between the balls and the globe is.
Now imagine both balls in the equatorial plane of the globe, one at zero degrees longitude and the other at the opposite side at 180 degrees. Imagine an observer at the equator at 90 or 270 degrees. Is this observer able to see one of the balls?
You got a proof now that two satellites are not sufficient for radio-communication around the Earth. we know the minimum could not be 2.
Now we think of three balls in in the equatorial plane of the globe. One at 0 degree longitude, the next at 120 degrees and the third at 240 degrees. The places at the equator exactly between two satellites are at 60 , 180 and 300 degrees longitude. We imagine an observer at one of these places, he is able to see two of the three satellites, so these two satellites are able to see each other too. Repeat this for all three places. ( If the observer on Earth is not able to any satellite, just increase the height of the sats above ground. You may use the globe for a test, is it possible to see both the 120 and 240 markings simultaneously? But a geostationary orbit is very high anyway.)
One satellite A is able to see two other ones, B and C. So also C does see both B and A. A is able to send a message to B, B can pass this message to C and C can send the same message to A. So a message may take one full loop around the globe. Radio-communication all around the Earth is possible using only three satellites.
But what about an observer at the north or south pole of the Earth? He never sees any one of the three satellites. But it was not required that the radio-communication all around the Earth using only three satellites should be useable from any place on the Earth.