Learn Russian: Пусть ферментация продолжается, и сыр созреет спустя три месяца. - Let the fermentation continue, and the cheese will ripen three months later.

The preposition спустя governs the whole phrase три месяца. Because три is a number and the object is inanimate, its accusative form is exactly the same as its nominative form. The word месяца is in the genitive singular because it is ruled by the number три, not directly by спустя. So, the entire phrase is structurally in the accusative case.

Could I use через три месяца instead of спустя три месяца here? What is the difference?

Yes, they mean the same thing here. However, спустя is slightly more formal and is most frequently used when narrating past events (like "three months later, he returned"). In future contexts like this cheese ripening, через is much more common in everyday speech. Additionally, спустя has the unique ability to be placed after the time period (три месяца спустя), which через cannot do.

Why is продолжается reflexive here instead of just using продолжает?

You use the reflexive продолжаться when the action happens on its own without a direct object (an intransitive action). If you used the non-reflexive продолжает, you would need to specify what someone is continuing. Because ферментация (fermentation) is the subject and it is continuing by itself, the reflexive -ся is required.

How does пусть work to mean "let" here? Does it change the verb form?