A derivative is basically how a small addition to the input value changes the output value; this change in output value is also studied in terms of being something “additional” (though it could also be a subtraction).

In L-calculus this approach is altered: how does multiplying the input value by a tiny value greater than one change the amount by which the output value gets multiplied?

Here’s the relevant Wikipedia page to get you started down the rabbit-hole: https://en.wikipedia.org/wiki/Multiplicative_calculus?wprov=...

An integral can be thought of as the infinite sum of infinitesimal areas below the graph (look up Riemann sums). I think what he is saying is that instead of summing these, you instead to the product. That is, multiply each infinitesimal quantity together.