Is this how logical operations and a computer work?
Questions and answers need to use the correct words, otherwise they are meaningless (or worse: misleading). 

A logical gate CAN be made from transistors. There are various other possibilities, from practical ones that have been used historically (relays, vacuum tubes) to esoteric ones (mechanical contraptions, cleverly formed molecules).

An AND gate produces at its output a signal level that is the AND function of its two (or more) inputs. The AND function (and the AND gate) are mathematical constructs. A real-world device can implement an AND gate, which means that it (within certain limitations) behaves according to the mathematical AND gate model.