To put a fine point on it: what DARPA did, was to sponsor a company called BB&N to develop a piece of hardware called the Interface Message Processor (or IMP). And that's pretty much it.

The IMP was the first gateway doing what you'd think of today as Network Address Translation, isolating "LAN" from "WAN" and using arbitrary computation to rewrite packets between the two. Though at the time, far more work was needed than just address translation. Wholesale network protocol translation was needed, as every site network (and there were already many small site networks) used its own networking equipment; and each vendor's networking equipment spoke some random stack of proprietary protocols invented by that equipment vendor. (There were nascent standards with open reference-impl hardware, e.g. MIT's Chaosnet, but none of these were widely adopted.) This was true all the way up to the application layer — different networking equipment required different application software that spoke the network's supported application protocols!

The IMP was a programmable router, allowing arbitrary CPU packet translation. So each site network could program the very same IMP with the details of its own network — what each type of local-network packet looked like, and what that should translate to for the WAN; and vice-versa.

This allowed these site networks to be glued together into a larger network. The IMP translated packets, and also "wrapped" each (proprietary, site-local) address of each LAN host, giving it a globally-routable name — i.e. an Internet Protocol address. This allowed machines on these networks to — at least in theory — address other networks' machines. All without anyone having to rip out any networking equipment, or replace each network's host application software with new software speaking standardized protocols.

Once the IMP was released, a bunch of universities and corporations came along and said to BB&N, "oh hey neat, I'll buy one of these! Heck, I'll buy one for each campus!" — and promptly stuck them into each of their (existing!) networks. (Some of these purchases were partially funded by DARPA as well — but only if the buyer reached out to ask.)

This didn't actually get anyone any value at first, because the IMPs still needed to be programmed, not just with the details of their local networking standards, but with the details of what the "WAN standard" application packet protocols would/should be for these local networks to translate things into. There were no standards for that yet.

So the folks doing the networking at these orgs, all got together to discuss how to actually get these boxes they bought to talk to each-other — e.g. what application-layer protocols they would need to invent/standardize on, to then get these gateway boxes to translate into from the proprietary site protocols they were using.

That group became known as the Internet Engineering Task Force, and their meeting notes became known as RFCs. (Read https://datatracker.ietf.org/doc/html/rfc1 if you don't believe me.)

Note that they called this WAN network formed by these sites through the IMPs the "ARPA Network" — presumably because that's what BB&N referred to it as, in turn because DARPA funded the IMP with the intent of creating such a network.

But DARPA had no involvement in the actual development of the "ARPA Network"! They weren't even a site on it! They didn't attend the IETF meetings! Rather, DARPA just kinda stepped back and said "go ahead, have fun" — and watched as the Internet took shape.

(I would thus describe DARPA's funding of the BB&N IMP as probably the most successful case of "nudge theory" in history. Almost as if someone at DARPA was a time-traveller who knew that that much effort, and no more, was all that was needed to shift the timeline.)