Question: I want the ability to animate a scene where I can draw boxes, circles, text, and other things like images, and be able to record it. I figure I will need my own recording software, but that is separate from this post.

Because my needs are very simple, the tool I use should be free.

Since some of the things will be computer science and math related, originally I was looking at 3Blue1Brown's videos, but the application used for those videos is a bit overkill for what I need.

Answer: 
[Godot Engine](https://godotengine.org/) is an open-source game engine and IDE that can produce games that run on multiple platforms, including the web. For the latter, it produces a WASM (WebAssembly) package that you can integrate with a few lines of HTML/JS into any web page.

Here's an example of a 2D game written in Godot, playable in the browser: [Wurst Day Ever](https://blinry.org/wurst-day-ever/).
The Unity web player still works, though it's no longer supported and requires the user to install it.

Supernova flash player is an option for legacy swf files.

Everything else is going to be HTML5, probably backed up with a canvas element. You can use plain javascript or one of the very many js libraries designed for this sort of thing. The first 3 I can think of which aren't already in another answer are phaser, melon and createjs.