The Douay-Rheims (http://www.drbo.org/drl/chapter/50001.htm) has:
The same is he that shall come after me, who is preferred before me: the latchet of whose shoe I am not worthy to loose.
Therefore "cujus ... corrigiam calceamenti" means "the latchet of whose shoe."
Cujus = 'whose' (Jesus's) and corrigiam = 'latchet' (object of solvam, 'I [may] loose[n]...
The New American Standard Bible was translated from the Greek (and Hebrew for the Old Testament), while consulting Aramaic texts, so the Vulgate doesn't really come into consideration. Here's the Greek:
Ἰωάννης μαρτυρεῖ περὶ αὐτοῦ, καὶ κέκραγεν λέγων, Οὗτος ἦν ὃν εἴπον, Ὁ
ὀπίσω μου ἐρχόμενος ἔμπροσθέν μου γέγονεν· ὅτι πρῶτός μου ἦν.
The word in ...
This seems to be about a rarer meaning of facere, namely "consider".
The verb habere can be used in the same way.
If I consider a house to be large, I can say: domum magnam habeo/facio.
(Or perhaps it is more common with a qualitative genitive?)
Thus ante me factus est means "he is esteemed before me", which is more naturally rendered in English as in the ...
The word his is a demonstrative pronoun in the dative plural form. It could be translated as to those. It corresponds to the word eis, which was translated as to them.
Therefore, the verse might be translated as:
But as many as received Him, to them he gave the power to become
children of God, to those who believe in His name.