What does the term affordability mean, with regards to economic
housing theory? Is there any economic reasoning for renting out a
property below the market rate within the social sector?

Affordability: "A consumer's capacity to afford a house. Affordability
is usually expressed in terms of the maximum price the consumer could
pay for a house, and be approved for the mortgage required to pay that
amount."

www.libertyhomeloans.com/glossary.htm

Experts tend to claim that a family should not spend more than about
30% on housing.  Affordability is usually used in terms of a
particular person/family and what they can afford for housing, but as
a general economic term it might have a different definition than I'm
aware of.

As far as charging less rent than the market equilibrium, the only
reason I can imagine is to keep tenants in the house.  Renting out a
house to new rentors is an expensive endeavor... if you have a realtor
or renting agency doing it for you then you're likely to spend 50% of
that months rent on their fees.  And that is in addition to not
receiving rent for the short while that the house is vacant.
By charging slightly less rent than the market will bear, the house is
more likely to stay rented and you're less likely to face the expenses
of getting new rentors.
There are of course downsides to charging a lower rent, but that's another topic.

It's a phrase often used by politicians at election time,when they
want to appeal to liberal voters or those who want cheap housing,they
never follow up on their promises so the phase is largely meaningless.