Let's mean let us as in Let us play dress up.
lets in this usage is a verb. He (Rafael Nadal) lets the other guys play dress up.

Rony is correct here. This style of writing is sometimes called headline style, because it omits words that are not necessary to understanding, especially articles and pronouns, as is common in newspaper headlines.

Also, it is in the present tense because it is a headline.

Surely it just means "Rafael Nadal lets the other guys play dress-up" which is perfect English for the third person singular present tense.

Or course!!!!

I hadn't even thought about that! I thought: let us the other guys .....no sense at all!

Thanks, everybody! I did think it was weird that a magazine would make such a stupid mistake...jejeje

I think (from your final response) that you now understand the construction but "just in case"...
The basic sense of "let" is permit/allow. this includes (by the way) the expression "apartment for let" which is not common in the U.S. (we would say "apartment for rent") but which I believe to be fairly common in GB (the sense being that the owner allows someone to live in the apartment, in exchange for the monthly rent).
The construction "let us" (or informally "let's") is what, in Latin grammar is called the "hortatory subjunctive" (one is exhorting others to do something). Most of the time in English one can't tell that it's subjunctive because for most verbs there is no difference. However, consider "Let's be friends!". It's the governed verb issubjunctive ("be") not the indicative ("are"). We also have, of course, simple indicative uses as well. Hence "A lets B do something." In this case "lets" is just the usual 3rd person, singular, present tense.