Learn Romanian: Am lăsat acești bocanci ai mei în fața ușii. - I left these boots of mine in front of the door.

Because the noun bocanci doesn't have a definite article at the end. When you put a demonstrative like acești before the noun, the noun stays bare. Without that definite article attached to the noun, the possessive mei needs the linking article ai to connect to it.

Could I drop the linking article and just say acești bocanci mei?

No, that is incorrect in Romanian. A possessive adjective usually needs a definite article nearby. If the noun has one attached, like in bocancii mei, you are fine. But if the noun is bare because of a word like acești, you must add a standalone linking article, making it acești bocanci ai mei.

Why do we use the linking article ai here instead of ale?

The linking article must agree in gender and number with the noun it refers to. Bocanci is a masculine plural noun, so it takes the masculine plural linking article ai. If the noun were feminine plural, like cizme, you would use ale, making it aceste cizme ale mele.

Why does the word ușă change to ușii at the end of the sentence?

The compound preposition în fața (in front of) requires the noun that follows it to be in the genitive case. The word ușă is a feminine noun, and its genitive definite form, which translates to of the door, is ușii.