The restrictions on grape products derive from the laws against using products of idolatry. Wine was commonly used in the rituals of all ancient religions, and wine was routinely sanctified for pagan purposes while it was being processed. For this reason, use of wines and other grape products made by non-Jews was prohibited. (Whole grapes are not a problem, nor are whole grapes in fruit cocktail).
For the most part, this rule only affects wine and grape juice. This becomes a concern with many fruit drinks or fruit-flavored drinks, which are often sweetened with grape juice. You may also notice that some baking powders are not kosher, because baking powder is sometimes made with cream of tartar, a by-product of wine making. All beer used to be kosher, but this is no longer the case because fruity beers made with grape products have become more common.
My mind immediately made the connection between this and "wine of your own make" as mentioned in Section 89 of D&C. I have a hunch that it's not a coincidence, but am not sure. I've also heard the Joseph Smith was concerned about or at least said he was concerned about "gentile" wine being poisoned, but don't know the source. Mainly, I'd like to hear if anyone has some good input or sources they can share about this.