That's an impersonal construction, which is used quite a bit in Italian but very little in other languages like English. Impersonal constructions are used to express ideas, actions or orders in a general way.
Examples of this construction:
Bisogna stare attenti (roughly: one must be careful/pay attention etc.)
Non si deve fare rumore (roughly: do not make noise, one should not make noise)
Qui si mangia bene (roughly: here one eats well)
Although these sentences don't have a real grammatical subject, they're always built with the verb conjugated in the third person singular, sometimes using the "si" particle (depending on the verb). That's the general idea.
With the verb "nascere", it's possible to build sentences like
Io nacqui povero
Io sono nato libero
But when we turn them into general impersonal statements, we use the plural:
Si nasce liberi
If we change the word order, the meaning becomes that of some quality or feature that is already present at birth, as opposed to being developed later:
Liberi si nasce (you are either already free at birth, or never)
Signori si nasce
So the example you cite translates roughly (but not literally) to "a (real) gentleman is born (not made)".
It's an idiomatic phrase, and as such it's not immediately explainable in a very logical way. In particular, I can't really tell you why in the impersonal formulation the plural is used; it's just the way it is.
The phrase is the title of an old Totò movie (a famous Italian comedian). In the movie he actually says:
Signori si nasce; e io lo nacqui, modestamente!
where the part after the semicolon, although strictly speaking grammatically correct (if a bit questionable), sounds quite peculiar and funny, which is of course part of the intended effect.