I'll take a shot at a straight answer.
When Parker O. Ackley first came up with idea of “improving” cases there weren’t nearly as many factory chamberings to choose from – the .222, .308 and their offspring and most of the belted magnums were still in the future. Existing chamberings generally tended toward those with a pronounced slope to the case, i.e. the .30-30, .250 Savage, 7x57 and so forth, so improving a cartridge to increase its velocity was a worthwhile goal. These days we probably don’t “need” an AI as much as 70 years ago since there are chamberings that split hairs that were already split two or three times over.
One thing AI’ing a case really does do is eliminate the need for trimming - I've fired AI cases 25 times and never had to trim them. Plus, they look cool and one can thrust out one's chest and casually toss off the name “So and So
Ackley Improved” when a member of the unwashed public range shooting clan asks you “what caliber izzat?”
T and A means tits and ass. See this thread:
.223AI Technical discussion, and only technical discussion 
