Is it normal to treat Math Theorems as "Black Boxes"

I think it's important to remember the key ideas in long proofs, as these may serve you later. Trying to remember every single equation in a long proof is a loss of time and energy. However, it is a good idea to memorize the statements of theorems in order to be able to recall them without having to reread them every single time. The thing is, the more you know by heart, the more you will be able to make links in your knowledge and recognize patterns.

I think one thing to bear in mind is that there is a big gap between "black box" and "learn in detail".

In my experience, people use the phrase "black box" for big pieces of machinery or big theorems, often from outside of their field, whose statements or conclusions they want to use but which they have little or no understanding of whatsoever.

Let's say "Learning in detail" means learning in a way that you could write out the details from memory, e.g. in an exam.

In between, let's say you are an analyst and you want to use some theorem about PDE or measure theory that you did not already know. Most of the time you don't do either of the two aforementioned things. You would look up the theorem in a textbook or paper and read enough of it to get the main ideas so that a) You could cite it precisely if someone asked and b) You could roughly explain the idea so that another practitioner in your field would ultimately find it very plausible.

In summary, this answer is similar to that of Guest

The question asked is refined --- albeit not definitively answered --- by Vladimir Voevodsky in recent works that include, in particular, the lectures "How I became interested in foundations of mathematics" (2014) and "UniMath" (2016).

Voevodsky's web pages provide links to both the lecture-slides (respectively, here and here) and the lecture-videos (respectively, here and here). These lectures are suited particularly to young mathematicians.

In a nutshell, Voevodsky argues in his "UniMath" lecture of 2016 that:

Today we face a problem that involves two difficult to satisfy conditions.

On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics.

On the other hand we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics.

The purpose of this answer is not to argue that Voevodsky's views are right or wrong in any absolute sense, but rather to suggest to anyone interested in these questions---students in particular---that Voevodsky's recent lectures and writings will amply reward careful study.