Learn French: J'ai répondu que mon petit-fils y allait souvent. - I answered that my grandson went there often.

The main verb J'ai répondu is in a past tense (passé composé). When reporting speech in the past, an original statement made in the present tense (He goes / Il y va) shifts backward to the imparfait (He went / Il y allait). This matches the sequence of tenses rule you are learning now.

Could I drop the word que here, since we often drop "that" in English?

No, in French you can never drop the conjunction que. While English allows "I answered my grandson went...", French absolutely requires que to connect the two clauses.

Why use the pronoun y instead of just saying allait là?

The pronoun y replaces a previously mentioned place (usually introduced by à, en, or dans). It is the most natural way to say "there" when referring back to a location, whereas là is generally used to physically point to a spot ("right there").

Where exactly does the pronoun y have to go in the sentence?

The pronoun y must be placed immediately before the conjugated verb it modifies. Here, it goes with the verb allait, so it is placed right in front of it to make y allait.