Learn Hardcore Hebrew: לא משנה אם נחזור באוטובוס או במונית, העיקר שלא נהיה עייפות מחר. - It doesn’t matter whether we go back by bus or by taxi; the main thing is that we won’t be tired tomorrow.

Literally, משנה comes from the verb לשנות / לשנות משהו in the sense of to make a difference / to matter here. So לא משנה means doesn’t matter.

Hebrew often leaves out an explicit subject in this kind of impersonal expression, so:

לא משנה אם... = It doesn’t matter if...
זה לא משנה אם... is also possible, but לא משנה אם... is more natural and common in everyday speech.

So the sentence starts in a very idiomatic way.

Why is it אם נחזור and not something like אם חוזרים?

Because the sentence is talking about a future possibility: whether we return by bus or by taxi.

In Hebrew, after אם meaning if / whether, you often use the future tense when the situation is in the future:

אם נחזור = if / whether we return
אם נלך = if / whether we go
אם נאחר = if / whether we’re late

So אם נחזור באוטובוס או במונית means whether we go back by bus or by taxi.

A present tense form like אם חוזרים would sound less appropriate here unless you were talking more generally or colloquially in a different kind of context.