It would be analogous in execution to PBS and NPR, in that it would hire professionals from the private sector but would focus on producing software for the public good instead of for profit.
In mission it would be more similar to the Interstate system, the point is just to give everyone good free tools, and then get out of the way and let commerce happen.
To some extent the open source movement is filling this niche, but as a mostly volunteer effort they lack the focus and accountability that a publicly funded organization would have. A Digital Infrastructure Foundation could prioritze usability and documentation in a way that the open source cannot (no-one volunteers to do things that aren't fun). The DIF could focus on just a few projects, instead of forking out to thousands of tiny efforts. One high quality image editor. One text editor, one 3D architectural program, etc.. All free, all open source. In fact the DIF could start with existing open source projects where appropriate.
There's a lot of thousand-dollar-software on the market today that sucks. If there was a good, supported, free alternative, companies like Adobe would be forced to innovate and compete. This is what Google did with search and email, and what they're trying to do with the browser, but Google as a company has a mission that includes making a profit. The DIF would have a mission of empowering citizens.
The US is (probably?) the leading exporter of creativity in the world; it is currently our competitive advantage. In order to maintain and expand this advantage, we need to get more and better tools into the hands of our citizens. Digital infrastructure is going to be nearly as important as physical infrastructure in the near future, and so it makes sense for government to start investing in this public good.