I keep hearing this phrase all the time. What does "now that that's out of the way" actually mean?
Referring to something as being "out of the way" means that it used to be undone or present and it is now done or no longer present.
"That's out of the way now." is a complete sentence using the phrase. We can substitute some reference to the thing itself for "that": "Filling up with gas is out of the way." or "We needed to fill up, and that's out of the way now."
"Now that..." is an introductory phrase indicating the current situation, as opposed to the situation in the past. "Now that you're finished filling up with gas, we can go".
Or we can use the two together, putting the two uses of "that' next to each other "Now that that's out of the way, we can go." There needs to be an antecedent for the second "that".