When I was a child, I learned that the term "out of" could be used to apply to a person or thing to describe where he, she or it was from. For example, a ship docked in Miami could be described as "out of Boston" as a means of describing that ship's hailing port, Boston. By the same token, a professional football player on the Denver team, playing in Dallas, might be described as being "out of Purdue" as a means of describing where he played college football. But now, I hear newsreaders say that a company is based "out of" a particular town. For example, they will say, "The Apex Corporation is based out of Philadelphia." This causes me to wonder, "Well, if they aren't based in Philadelphia, then where are they based?" Of course, the newsreader actually meant to say, "Apex Corporation is based in Philadelphia." As best as I can remember, the term "out of" came to be the preferred alternative to the word "in" sometime during the early 1980s, as I don't remember hearing this error prior to that. I'd be interested to know if there is a documented example of it appearing before that time. For those of us who learned the language long before the 1980s, this is especially confusing as the term "out of" means the exact opposite of the word "in." Why would someone use a two word combination (out of), that means the exact opposite of what they really meant to say, when a single word (in), that describes what they really do mean, is a better choice of words?
I think based out of is often usd to refer to organizations with a widespread presence. So they are not actually in any single place, that's just the location of their main headquarters. From there they've spread out to all their field offices and foreign subsidiaries.