How to “Understand” Calculus?

first you need to intuit how a cartesian coordinate system works so you understand how a graph works

differential calculus is about rate of change: simplest case means slope or rise over run, practical case means estimating the slope of the portion of a curve.

  This can be a real physical curve or it can be a curve defined by a plot of numeric values on a graph

integral calculus is about estimating the area defined by a curve this is a game of cutting the area up into pieces adding up all the regular pieces and estimating the little pieces that fill in the edges.

FWIW to you here is a wiki link re calculus:

https://en.wikipedia.org/wiki/Calculus

and cartesian coordinate:

https://en.wikipedia.org/wiki/Cartesian_coordinate_system

for cartesian coordinate graphs you should understand euclidian geometry:

https://en.wikipedia.org/wiki/Euclidean_geometry

and likewise euclidian geometry requires algebra:

https://en.wikipedia.org/wiki/Algebra

That's an awfully general question. I would expect any high school or 1st year college course or textbook to do a reasonably good job of providing you with the basic intuition behind derivatives and integrals. It would help if you could be more specific about exactly what it is you are trying to understand.