As you say, what you're missing out on here is how the three parts of the triangle work together. Each part has a stake in helping the other part out. Let's look at your example.
The bill that is being proposed would require the government to buy lots of new computer technology, right? So the interest groups want that because the computer technology and probably some expertise (like for programming and such) will be bought from them. At the same time, the bureaucratic agencies like it because they will be the ones getting their budgets increased to buy the technology and they will be the ones who get more "turf." Finally, the committee members can benefit because the interest group will give campaign donations to them and will perhaps run ads for them and endorse them. So everyone has some stake in the bill. Everyone gets something and they all have an incentive to work together. This can become a problem because, as in this case, no one has an incentive to try to save money or reduce government waste.