What is "like a bat out of hell"?
The phrase has to do with the speed something is done (fast)—like "Bob drove to the doughnut shop...like a bat out of hell!" (This, of course, assuming Bob has an almost unnatural predilection for deep-fried tori—perhaps he's fiending for a glazed...) (Well, you know Bob....) So, more specifically, it isn't just "fast" but that the aspiring chiropteran in question is going somewhere. One might type or bake cookies or do laundry like a bat out of hell but I doubt it. It seems more at home with driving, running, and flying (of course). A bat out of hell should be going somewhere.
Whence comes "like a bat out of hell"?
The first recorded usage of the phrase, in print, dates from 1921. As is common when dating a word that way, the usage almost certainly predated that—perhaps by several years. And therein lies the probable source. Bats are very quick fliers (if you aren't quick, you fall out of the sky— you know how fast those things are). This is necessary due to many being insectivorous. Additionally, the bat manages to do so in primarily dim or almost no light; a feat made possible by the development of a biological sonar (often incorrectly described as "radar") called echolocation. On top of all that, the whole "scary" appearance thing—there was a reason Batman chose the creature as his symbol/costume, since, you know
Criminals are a superstitious cowardly lot. So my disguise must be able to strike terror into their hearts. I must become a creature of the night, black, terrible...a...a...a bat!
So we have a speedy little, acrobatic, "scary" mammal who can maneuver at high speed even in the dark. Seems a great description of WWI flying aces, doesn't it? Well, perhaps not. But some must have thought so. The phrase became a way of describing aviators (maybe an aviatrix or two, but this was the early twentieth century when few women were allowed to do anything like anything that might be "coming out of hell") going at top speed. (Whether doughnuts were involved seems unclear.)
As for the latter part of the phrase, it's just a colorful addition to what would otherwise be a boring simile (according to my source, it "probably refers to the bat being 'from hell,' not necessarily trying to leave hell" Okay...). Think of it this way: "Like a bat." So what? Who cares? Ah! But like a bat out of hell—now that's something.
Finally:
We say some guy was "goin' like a bat outta Hell." How do we know how fast a bat would leave Hell? Maybe he would leave real slow. In fact, why should we assume that a bat would even want to leave Hell? Maybe he likes it there. Maybe Hell is just right for the bat. Maybe it's bat Heaven. And now that we're on this subject, how do we know Hell has bats in the first place? What would a bat be doin' in Hell? Usually a bat is in the belfry. Why would he want to split his time between two places? Then again, maybe that's why he's in such a hurry to leave Hell. He's due back at the belfry.
George Carlin, Brain Droppings 1998.
(Sources: www.word-detective.com/122099.html#batoutofhell; Batman quote from Detective Comics 33 (November 1939), scan of the page from members.ttlc.net/~bobhughes/BobKane.htm)