Honestly there is a truth and a super-truth, to use theater terms.
The truth, is that you should look at the Broad Agency Announcements (BAAs) that DARPA, specifically the Information Innovation Office (I2O) makes. There will be big PDFs that have lots of boiler plate and look really boring, but these are basically documents written by a program manager (PM) that describes what their envisioned research program is, how they see it being divided up, and what kind of researchers they see doing work in each area of their research program. A research program is split up into one of many (between 3 and 8) technical areas (TA) and there will be language in the BAA describing how many TAs one performer might propose to.
Overall a TA can be described at a high level as one of a few roles: a blue team, a red team, and a white team. There will usually be more than one blue team, at most one red team, and precisely one white team. A blue team does the R&D on the program. Each blue team can have their own unique approach to addressing the research challenges laid out by the PM. Usually the blue teams have competing or complimentary approaches, it is seen as a waste of money to pay two blue teams to do exactly the same thing. A red team evaluates the work done by the blue team. Exactly what this means depends on what the blue team builds. The white team "holds the room together" by fitting the PMs vision together with what the blue team does. The white team usually has a much closer relationship with the PM and usually the white team is pre-determined by the time the BAA is made public.
Blue teams are usually teams of 2-4 companies / universities. These teams usually form before a BAA comes out due to people "in the know" getting a rumor about an upcoming program, or just after it comes out, based on prior relationships. It's kind of rare to have a blue team that is only one entity, but depending on the entity and the size of the program, it can happen.
The path to a successful proposal is demonstrating that your team has ideas that are relevant to the PM and that you can successfully execute them. You show this by having a stack of prior work, good ideas, and a well formed and coherent proposal. Think of the proposal as an audition - if you as a team can't get it together for a month and write a 30 page document describing what you want to do, you probably can't keep it together for 4 years working on something. Your team reads the BAA, does some analysis, figures out how your ideas and past work can apply to the research program proposed by the BAA, and then you write the story up. If you do a good job, and your story is more compelling than other people, you get the money.
The super-truth is that it's an old boys club with a lot of luck and nepotism.
The way people get "in" is when their peers go to DARPA to be a PM. Then you, the researcher, can think "Well Dr. Smith knows about X and likes X+Y, I do some Y, I have an idea that could use some funding, what would Dr. Smith like to read about." Then you write it up and e-mail it to them.
Luck is a big factor here because PMs get e-mails like this all the time. It's almost essential to already be in the PMs rolodex/friend group to get some of their cycles. They like to say that they want to have people outside their circle approach them with new ideas but that's kind of a white lie, there are lots of people in the world and a finite amount of the PMs time.