No le puedo mostrar este video a nadie.

What role is “le” playing in this scentence?

The le (a personal pronoun) in this sentence indicates to the listener that there’s an indirect object coming up later in the sentence - in this case, a nadie. Having both le and a nadie in the same sentence is a bit redundant, however, if the indirect object is already been established in the conversation there’s no need to repeat it.

The real usefulness of these pronouns is when you start combining direct object and indirect objects and don’t want to repeat yourself endlessly. For example, you could say “No puedo mostrarsélo” to convey the same meaning as the cloze sentence assuming the identities of the pronouns is already known.