When I first started seeing 当店 I wondered what that was as I had never seen 当 used for ”this" in any of my textbooks. I always associated it with food. 弁当. 
Aside from the often used, the kanji saves print space explanation, why? Is このみせ wrong? Would one ever say このてん? Anyone know the etymology of 当 being used as “this”? And where you might use it instead of この?
当店 is a more formal expression. It would give a sign in a store that was intended to be professional or serious a different feeling if you used another way to describe it. You probably would not say it yourself.
この店 is not wrong, though note that it’s common to add the honorific お to 店 when you are talking about a store, so you would get このお店.
このてん is not something you would say to mean “this store”.
Thank you much. Indeed I see it on signs and websites of retail and commercial (当社) establishments.
Interestingly enough it can also be used more like “that” in 当時 or maybe even 当人
Henshall says that it’s a much-simplified version of 當, which “originally referred to offering a field as surety for a loan, the amount of land to be pledged being determined in proportion to the sum advanced and being therefore deemed appropriate”. All the other meanings including both “hit” and “this” accumulated by association.
I might be wrong on this, so someone with more Japanese experience feel free to correct me, but I believe 当 is not only simply more formal, but it also requires context. Whereas この店 might just mean “this shop right here” and may introduce it as a new topic, 当店 is referring to the shop that was mentioned in conversation.
Hence I always remembered 当 as “said” instead of “this” or “that”, e.g. 当店=said shop, 当人=said person, 当社=said company etc.
In Chinese, 当 is actually a simplified version of the traditional character 當. As a verb, they mean something along the lines of “to act as”, but there are also words such as 当时 (trad. 當時) with basically the same meaning and usage as in Japanese.
It doesn’t have to be already mentioned in conversation. If you are in a store then a sign might say 当店.
As another example, this online bakery’s page for this item describes it as “
当店人気No.1セット ねこねこ食パン”, as basically the first text on the page.