I'll try to give a slightly different perspective on the same things covered by the other answer.
A "table of truth" characterises a gate by telling you how each basis state evolves through the gate.
Note that this requires choosing an input and and output bases.
In the prototypical example of the CNOT gate, one chooses the computational basis both for input and output states, and it turns out that all the elements of the computational basis evolve into other elements of the computational basis.
In other words, an element of the computational basis, passing through a CNOT gate, ends up in a specific output basis state (as opposed to a superposition of basis states).
What might be confusing in this "table of truth" way to describe a gate, is that it can be used only with some choices of input and output bases.
For example, you cannot give a "table of truth" description of the CNOT gate using the $\{|L\rangle, |R\rangle\}$ basis, because, as you can check, $\text{CNOT}|L, L\rangle = \frac{1}{2}(|L, L\rangle + i |L, R\rangle + |R, L\rangle -i|R, R\rangle)$.
Indeed, the "table of truth representation" is only useful in some circumstances, for example when one wants to highlight that a gate might be a "quantum generalisation" of a specific classical gate, like it's the case for the CNOT gate.