Peter, fastening his eyes upon him, with John, said

Both Peter and John looked at the man, but only Peter spoke.

fastening his eyes upon him

Possible meanings are 1) "looking directly at him" or 2) "looking intently at him" (See: rc://en/ta/man/translate/figs-idiom)

Here the word "looked" means to pay attention to something. AT: "The lame man paid close attention to them"

It is understood that Peter has the ability to heal the man. (See: rc://en/ta/man/translate/figs-explicit)

Here the word "name" refers to power and authority. AT: "With the authority of Jesus Christ" (See: rc://en/ta/man/translate/figs-metonymy)