I'm a little fuzzy with your description above as to whether or not you are living with her or not. If you moved into her place and she is living someplace else, but still has things in this house that she wants to keep, I'd probably do my best to honor her wishes. However, I'd box everything in the pantry up and put it in the garage. And since you need to use the fridge, then maybe ask her to move what she wants to keep to her place and toss the rest. If she's living with you, then ask for a couple shelves in the fridge for just your stuff and make sure you make and dish up all the meals for you and your family.
If she lives elsewhere, I'd just make sure you pack food/snacks/drinks for your kids if she is to watch them. If you don't trust that she'll use only your food, then I wouldn't bring the kids over w/out parental supervision.
I don't think having a talk will fix this and just throwing things away, since they don't really belong to you doesn't work either. I have somebody in my family who hoards things and saves way old food. I know it's a generational thing. If you didn't grow up with much and you didn't ever waste a thing, it's a hard thing to shake...throwing away graham crackers was probably unheard of. As a parent, your job is to make it clear how things go with your kids, if she can't do that for your kids, then she doesn't get to be alone with them.
The person in my family who is this way... well, I don't like that the fridge is filled with sour cream that is 3 years old, but it's their fridge and their choice, but when I visit I just don't eat at their house. And I pack food for the kids.
Good luck!