My 7 year-old once asked me "What's calculus?".
I've never liked the approach of lying to a child because it is believed they would not understand the true answer, and so telling them something that's false. So I thought for a bit and came to this explanation.
My son and I had been talking about fractions a lot, and had been drawing lots of number lines. To give the concept some context, I explained that fractions were numbers that allowed us to talk about "parts of things". Without fractions, using only integers, the best we would be able to do is to say things like "some" or "a little bit of". So fractions were a new kind of number that allow us to talk about the exact amount when dealing with parts of things.
To answer his question about calculus, I drew a number line and marked off 1, 2, 3..., then asked him to mark the number that comes right after 1. Since we'd been talking of fractions, he didn't go for 2, but instead marked right beside 1. I circled that mark and drew a call-out to "zoom in" and show that, between those two marks was more space, that they weren't right next to each other after all. It took only 1 or 2 more times of zooming in for him to catch on that we could play this game forever.
I explained that calculus was a way of creating a new kind of number that allows us to talk about "the real number that comes right after". This new number is a differential, dx.
Integration is simply a "sum" of differentials, differentials generated by a differentiated function.