Here are the definitions that require clarification:
Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true.
Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false.
Consider logical consistency first using two sentences symbolized as P and P V Q where "V" means "or". If we allow P and Q to range over the values true and false is it possible for some valuation to make both of the sentences in the set true? We can find that out by putting the sentences in a truth table generator joining them with "and" represented by "&":
Note that there are two different valuations of P and Q that will make both of the sentences true. By the definition we only need one valuation so we have more than we need and the set of sentences are consistent.
This set of sentences, by itself, is not an argument. An argument has a set of sentences, called premises, and a sentence called a conclusion. Suppose we make the argument that the set of sentences P and P V Q entail the sentence Q.
Without considering differences between entailment and validity, we could put that in a truth table generator to see what would happen:
Now consider the definition. There are two ways for the set of sentences to be true. We only needed to have one of them for consistency, but we need to consider both for entailment. In both of those cases the conclusion must not be false. However, it is false when P is true and Q is false. So logical entailment fails.
Consider the final question:
I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument).
Without knowing precisely how the textbook distinguishes between logical entailment and logical validity, it is possible to entail a conclusion from no premises at all. Such a conclusion would have to always be true. Since the conclusion is never false, it is impossible to have a set (even an empty set) of true premises and a false conclusion.
An example would be P V ~P. This truth table shows that this sentence is always true:
Such sentences are called tautologies. They are always true.
Michael Rieppel. Truth Table Generator. Retrieved on July 12, 2019 at https://mrieppel.net/prog/truthtable.html