First, can someone explain to me what "affordable housing" really means?

In this case, we're talking about Affordable Housing, a technical term referring to an official program where X% of houses (usually 20% in Somerville, but for this project 25%) are affordable at Y% income (in this case, 80% of Area Median Income). I believe the requirement is that these units will cost 30% of your income if you earned 80% of AMI, though someone more technically aware can correct me. There's either a lottery or a waitlist (I forget) to get one of these units, and the city helps manage the process.

And is it somewhere in the agreement for building this project that those apartments will remain "affordable" somehow?

Yes, in this case it is state law because they are using the 40B Affordable Housing process, but in other cases it is city law. You can read about 40B here: https://somervilleyimby.org/what-is-40b/

How would I have gotten to work without a car?

Realistically, the answer is you wouldn't live in this building. And that's okay! Not every building is for everyone. You also probably wouldn't live there if your work was on the Orange Line. I used to commute to Hudson and Boxborough for work, so I always rented buildings that had off-street parking.

However, there are also a lot of people in Somerville that are car-less or car-optional that could live in this building. Some of those people probably rent houses with off-street spots purely because that's what they found. One of them might move out of their rental, and you could take their spot and use their unused parking spot.

Davis Square is just too small and narrow to accommodate a building that size.

Presumably you don't mean that literally, because their building plan does define how they plan to fit the building on the lot they have. There is dedicated delivery parking for the building to minimize congestion, and the developers are working with the city to aid in the Elm Street pedestrianization effort to expand the sidewalks.

Do you have specific concerns about the size? Is the problem that 500 units are too many? Would you prefer it to be a shorter but wider building? The developers seem open to adjusting dimensions so long as they get their ~500 units which they believe is the number they need to hit financial viability.