I'll give you a fast-forward of the subjects you need to study to actually learn this stuff (with links). Many of these are part of typical computer science curriculums. However, most will exclude at least some of this to its fullest. Note that it's a lot of knowledge, you might find it frustrating to just deep dive into it on your own, don't take it the wrong way, maybe follow an actual course and you will be guided into all of this. Maybe I'll also try to give a run-down, skipping the preupedeitic knowledge... later... if I have time.

P.s.: to get the general feeling you don't really need to know all of this tho. Just skim it. And dive deeper if you like it.

Any respectable computer scientist needs to know at least some first order logic https://en.wikipedia.org/wiki/First-order_logic
- Read the semantics part. You don't need everything, but you must understand what it means for a Formula to be a logic consequence of a set of formulas
- Read a bit the deductive systems part. I find "natural deduction" (in detail here https://en.wikipedia.org/wiki/Natural_deduction) easier to understand (more... Natural) however, your mileage may vary, many people find it confusing, hence why "sequent calculus" was invented. We'll use the sequent calculus later
You need to know lambda calculus https://en.wikipedia.org/wiki/Lambda_calculus If you know any functional programming language it will be easier, but it's optional
- You can learn it on its own, but it fits better as part of a "foundations of computer science" course. I don't have a link for that.
You also need to know a bit of programming in basically any strongly-typed language. This will give you some kind of general idea of what types and type systems are.
Now you can put together what you know of types and sequent calculus, and learn what type derivation rule are https://en.wikipedia.org/wiki/Typing_rule
- Now you understand the meme
- You can also check out what type systems are as a whole https://en.wikipedia.org/wiki/Type_system
- We have skipped Curry-Howard, but you can see how typing rules look a lot like logic deduction rules, in particular sequent calculus
Hooray, time for Curry-Howard https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence they basically notice the same similarity you just noted, but take it a step further, they can do it because they know other logics other than first order logic
The next step, if you are interested, are theorem provers, which are the direct application of the correspondence https://en.wikipedia.org/wiki/Proof_assistant
- Check out rocq or agda