Arthur Andersen was the largest and most successful accounting firms, as such they had deep business contacts with basically the entire business world.
As mainframes moved from the white glove ivory tower to more access for programmers, then to minicomputers, then especially in the microcomputer revolution of the 1990s, Arthur Andersen was well connected to sell IT and development services to a very very large range of clients. This wing was "Andersen Consulting".
Andersen Consulting's partners got tired of taking all these sweet revenues and shoveling them to the tax partners of Arthur Andersen, so they split off in 1998 or 1997 or so, and became Accenture.
Also around this time, Accenture signed many deals to outsource entire IT departments to them. Stupid companies not seeing IT as one of the primary efficiency and market advantages (see:Amazon) but instead as an annoying cost center jumped at the chance. This made them beholden to Accenture.
Anyway, Accenture had the salespeople and contacts, the size, and the reputation of being able to deliver large projects, or they'd stick them out. Would they be great? No. But they'd stick it out.
All the big 6: CoopersLybrand, Ernst and Young, Deloitte, Price Waterhouse, KPMG, were all in on it at this point. But Accenture had the largest partner-to-workerbee ratio, so they raked in the cash. Accenture had the ability to hire armies of workers, rarely from computer science backgrounds, crash course them in programming over 6 weeks, and then shuttle them off to be vastly overpriced consultants. They even had their own private college for this (St. Charles).
Basically, they had contacts and a good enough pipeline to create worker bee programmers for large projects.
The partners generally came up the ranks so they could speak the IT lingo, and they were promoted/selected based on ability to produce revenue and sales. With the huge size and contacts and labor pool, plus happily jumping into any and all software implementation.
Accenture and their ilk were also key for supplying huge numbers of people for SAP and other package implementations. Basically, these software companies couldn't muster the armies of implementers necessary to install and customize their software. Accenture and their ilk could. These ERP packages likely helped fuel the productivity boom (all of it went to the rich) of the Clinton era as computers went from 10s of MHz to gigahertz processers as well as memory and storage increases and the internet boomed.
You probably can't imagine an executive management team that is ignorant of IT and IT practices, and of course the old enterprises have plenty of these. These execs just wanted to hand off these annoying things to someone, and Accenture was there for them.
Modern executives need to be IT-aware. I'd say this is only very very very recent, with the rise of Amazon. Before then you'd have the CEO that proudly annouced they don't even know how to use Email.
Those CEOs are a dying breed. Your orgs will live and die by navigating the IT landscape. Amazon sucks for many reasons, but they do a lot right, and one of those things is "don't use consultants, build it in-house".