In "Я подъезжаю к магазину" it's used because the person in question approaches the store in his/her car, and that's exactly what s/he wants to tell you. We don't even know if this store is the main destination of this trip (is he going to buy something? is he meeting someone near the store? or just passing by?). The store is just a convenient reference point.
Yep. I would probably translate this sentence as something like "I am driving towards the store" or "I'm in my car and heading in the general direction of the store", etc. (Note that the implied meaning of "moving towards; approaching" comes from both the verb prefix под- and the preposition к + dative.)

But by the way, к + dative can also be used with a person (rather than a building or a city or country, etc.) in the dative, and then it's often understood as meaning "to someone's home". Мы едем к бабушке ("We are driving to Grandma's house"). Я иду к Ивану ("I'm walking to Ivan's place"). You can also invite someone else к мне ("to my place; to my house"), but as I said above, if you're going to your own home, then it's домой without a preposition.