What is the branch of mathematics called that deals with proofs? What is the branch of mathematics which deals with how mathematical proofs are constructed?
I am looking to learn more about cryptography and category theory, but I am missing some of the mental tools which I need.  In particular, authors often just state things as true, and I cannot see why they are.
In particular I am very rusty in what was called "pure maths" at A-level in the UK.
 A: Writing this as an answer as I have exceeded the comment character limit.
Gathering the comments made so far, I believe that "Proof Theory" as specified by @KevinQuirin is what you are looking for as a direct answer to the title question. 
However, if you want to learn more about cryptography and category theory, you should check out books that are relevant to those fields and appropriate for your skill level. A lot of them tell you what the pre-requisites are for studying the content in the introduction(this is usually the case in most mathematics textbooks). 
Moving on to your point of "authors often just state things as true, and I cannot see why they are." A lot of mathematics builds upon itself, and some books don't go through every step to show you exactly how every proof has been derived. I've completed A-level mathematics not too long ago, and I have gone through textbooks that just say "it is obvious that" and then state an answer. This can be dealt with if you are able to prove whatever is asked of you, step by step, from a point which you are certain is true.
Proofs are generally derived by assuming a set of axioms are true and building upon them. I've got quite a basic example that proves the quadratic formula. I explained this to my cousin who just started his A-level mathematics course.The book just got him to use the quadratic formula as it assumed him to know how it was derived. It assumed that everyone doing A-level maths knew how to prove it on their own. He did not understand what the term discriminant meant either. This led to further confusion when he attempted exercises from the textbook he was using.
Example I showed him:
$ax^2+bx+c=0 \Rightarrow x^2+\frac{b}{a}x=-\frac{c}{a}$
Completing the square:
$(x+\frac{b}{2a})^2=-\frac{c}{a}+\frac{b^2}{4a^2} \Rightarrow (x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2} \Rightarrow x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a} \Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Where $b^2-4ac$ is the discriminant.
This cleared up the entire topic for him. And it is a very basic example of what I mean by building from a set of axioms which you know are true. All I used was some surds and basic algebra.
Most undergrad-level books -which I assume the level cryptography and category theory books are at- assume at least an A-level understanding of pure mathematics. So my advice would be to brush up on that using a good textbook (I recommend Mathematics, The Core Course for A-level by L.Bostock and S.Chandler) and then tackling a book on cryptography/category theory at the level which you want to study them.
