Learn Hardcore Japanese: syanpuu no tumekae wo kai ni ittara, senzai mo tokubai desita. - When I went to buy a shampoo refill, detergent was also on sale.

In シャンプーの詰め替え, it refers to a refill pack of shampoo, not a new full bottle. In Japan, many products are sold this way, so 詰め替え is a very common word on packaging.

It comes from the verb 詰め替える, which means to refill or to transfer into another container.

Why is there a の in シャンプーの詰め替え?

The の links the two nouns and shows their relationship.

So:

シャンプー = shampoo
詰め替え = refill
シャンプーの詰め替え = a shampoo refill / refill for shampoo

You can think of の here as something like of or for, depending on what sounds natural in English.

Why is it 買いに行った and not 買うに行った?

Because with the pattern verb stem + に行く, Japanese uses the stem of the verb, not the dictionary form.

For 買う:

dictionary form: 買う
stem: 買い

So:

買いに行く = to go to buy
買いに行った = went to buy

This is a very common pattern:

食べに行く = go to eat
見に行く = go to see/watch
飲みに行く = go to drink