Fan-out describes a signal that feeds more than one gate. If input B goes to both an AND gate and an XOR gate, B has a fan-out of 2.
For circuit analysis the rule is simple: a fanned-out signal is one value. Whatever B is, it is the same at every gate it reaches. When you evaluate a row of the truth table, write B's value once and use it everywhere.
Fan-out often ties a circuit's conditions together and hides a simpler function. For example, if A feeds both an OR and an AND as , the absorption law reduces the whole thing to just A.
In real hardware, fan-out has a cost. Every input a gate drives adds a little electrical load, which slows the signal's rising and falling edges. Each logic family therefore has a maximum fan-out, and designers add buffers when one signal must drive many inputs. Large fan-out also adds to propagation delay.
Don't confuse it with fan in, which is the number of inputs a single gate has.