Translation:The Szabó family is moving away from their old street.
The English would be more elegant if the nouns, pronouns, and verbs matched up regarding number:
1. The Szabo family moves away from its old street.
2. The Szabos move away from their old street.
?3. The Szabo family move away from their old street.
Is #3 grammatical? It is if you construe the collective noun 'family' as plural. British English often construes collective nouns as plural, whereas historically American Emglish usually does not. However, nowadays American English also sometimes treats collective nouns as plural.
... American English also sometimes treats collective nouns as plural...
-------- i keep hoping the trend will die out - but with soccer's (futbol ) increasing popularity and football's increasing instances of traumatic brain injury, we hear more and more "the Liverpool team are .. . " . . .
Big 12 sep 18
I believe you would say what DL says here at the top of the page:
"Szabóék elköltöznek a régi utcájukból."
In other words, I think you could translate "Szabóék" either as "the Szabos" or as "the Szabo family". The literal Hungarian word for family, "család", does not appear in the Hungarian sentence we are given.