The word "few" unitizes the time reference, turning it into something you are counting (few means a small number; it isn't a specific number but it still means that you're counting months). As a countable reference, it needs an article. The word "following" means that it specifically refers to the next period, not some random months occurring at an unknown future time, so "the" is the appropriate article.

Without "following" you would need some term to anchor it in time so the sentence is meaningful. If it said something like, "This problem became evident during a few months that summer, you could use "a" because it wouldn't be a specific reference.

Without "few", the term "following months" can be an open-ended reference to non-countable time, with months only suggesting an appropriate size measure for the scope. In that case, you wouldn't necessarily need an article: "This problem became evident during following months." "The" would be optional.

The word "few" unitizes the time reference, turning it into something you are counting (few means a small number; it isn't a specific number but it still means that you're counting months). As a countable reference, it needs an article. The word "following" means that it specifically refers to the next period, not some random months occurring at an unknown future time, so "the" is the appropriate article.

Without "following" you would need some term to anchor it in time so the sentence is meaningful. If it said something like, "This problem became evident during a few months that summer", you could use "a" because it wouldn't be a specific reference.

Without "few", the term "following months" can be an open-ended reference to non-countable time, with months only suggesting an appropriate size measure for the scope. In that case, you wouldn't necessarily need an article: "This problem became evident during following months." "The" would be optional.