Learn French: Pendant l'affinage, la croûte va en durcissant. - During the ripening, the rind keeps hardening.

The structure aller + en + present participle (the gérondif) is used to describe an action that evolves progressively. It often translates to "keeps doing [X]" or "gets more and more [X]." Here, va en durcissant highlights that the hardening is a gradual, ongoing process throughout the ripening.

Could I just say la croûte continue de durcir?

You could, but the meaning shifts slightly. Continue de durcir simply states the fact that the action doesn't stop. Va en durcissant focuses on the progressive intensity—it tells us that the rind is getting harder and harder as time passes.

Why does the present participle of durcir have a double 's' (durcissant)?

Because durcir (to harden) is a regular second-group verb ending in -ir. Just like finir (which becomes finissant), it takes the -iss- infix in its plural present tense forms and in its present participle before adding the -ant ending.

What kind of ripening does l'affinage usually refer to?

In French food culture, l'affinage is the specific term for the aging or maturing process of cheese (le fromage). It can also be used for curing meats or refining oysters, but a French person reading this sentence will almost certainly picture a wheel of cheese!