Learn Spanish (Latin America): Subo al taller a que me ensanchen y lustren las botas. - I go up to the workshop to have my boots stretched and polished.

Because the subject changes. If you were going up to stretch the boots yourself, you would use a plus the infinitive, like subo a ensanchar. But since you are going up so that someone else does the work, you must use a que followed by the subjunctive verbs ensanchen and lustren.

Could I use para que instead of a que in this sentence?

You could, and it would not be grammatically wrong, but a que is the most natural and standard choice after verbs of motion like subir, ir, or venir. It neatly combines the destination preposition a with the purpose marker que.

Why does the English translation say "to have my boots stretched" when there is no verb for "have" in the Spanish sentence?

English uses the causative structure "to have something done" when a professional performs a service for us. Spanish handles this simply by making the unnamed workers the subject of a third-person plural verb. Literally, the Spanish phrasing means "so that they stretch and polish the boots for me."

Why does the sentence use me and las botas instead of just saying mis botas?