The math videos creator 3blue1brown has a webpage and in the FAQ section he goes into detail about how we went about creating his videos. Here's his answer for the question "What do you use to animate your videos?"
I create most of the animations programmatically, using a python
library named "manim" that I've been building up. I’ve also used
You could try VPython (https://vpython.org/). It's free and does 3D animation. It was built for teaching physics, and can visualize vector fields, graphs, various primitive objects, etc. as well as animating them, and letting you interact with them.
There are many good options for the "presentation" one. If you are doing multivariate calc stuff, then CalcPlot3D is a great option. (I also recommend many others such as Sage Math, which created this amazing graphic using Tachyon for image production, but CP3D is the easiest to start with.)