"What time does this store open?"
OK I'm bad at explaining but I'll try. So basically に is like a to (go) so I'm guessing it's like this shop what time is it /going to/ open. English isn't my first language so I'm bad at explaining things. Like what time is leading to opening.....??? OK I should stop I'm gonna be more confusing
開ける (akeru) is a transitive verb, which means it can take a direct object. Direct objects are marked with を. ドアを開けます (doa o akemasu) - I open the door.
開く (aku) is an intransitive verb, so it can't take a direct object. You can't say ドアを with this verb, so "door" becomes the subject and you say ドアが開きます (doa ga akimasu) - the door opens.
In this case, saying that a store opens is using the verb intransitively. The store is not the direct object but the subject.
I often think of it like "the door cannot open something" for transitive. In these cases we need the intransitive verb.
Or put in an example: (in a horror movie) ドアが開きます (but you don't know who opened it).
ドアを開けます and it makes a rusty sound (referring to the horror movie)
The kanji is the same and the meaning is the same verb “open”, but what is the difference? To begin with, let’s look at “hiraku”. Please imagine when a flower blooms. The bloom opens from the center and outward. This has the feeling of gradually being able to see the center. It is moving toward more than two angles.
Next, “aku” is to open by moving a thing like a key or lid that blocked a passageway. And, the movement is in one direction.
About this question specifically:
mise o hiraku: Open a store. (meaning “open a new store, start a business”)