The basic idea of a proof is you chain together a sequence of logical implications. How do you know a proof is correct? There is a definitive way. Hopefully the following helps.
Consider:
Statement $P$: $p$ is a prime number larger than $2$
Statement $Q$: $p$ is an odd number.
By writing $P\implies {Q}$, you are saying "if $P$ is true, then $Q$" is true.
Proving this statement requires basic knowledge of prime numbers and isn't difficult.
In a long proof, such as proving $\pi$ is transcendental, you are chaining together say dozens of implications via sound arguments.
$P_1\implies P_2 \implies ... \implies P_n$, where each pair requires it's own reasoning.
There is no inherit difference between how proofs are done in any field of math. The trick is how do you chain the statements together? If all links in the chain are logically valid, then the proof is correct. That is the only criteria needed. The rest boils down to taste/preference. For example, maybe there is a shorter proof where certain hypotheses can be relaxed. This is a matter of style, but never influences whether or not the proof is correct or not.
This requires any combination of the following three:
$(1)$. Technical expertise
A student, now matter how brilliant and talented at logical thinking, is not going to prove a research level theorem about compact operators in Hilbert Spaces by only knowing calculus theorems. This is knowledge, and the more you know, the more resources you can pool from to prove a theorem.
$(2)$ Tricks of the trade
Here I refer to common methods/themes of proving theorems in a field of math that are not immediately obvious to newcomers, but most experts would attempt a proof using these tricks.
An example from PDEs: Prove there is a unique solution to a given PDE. The common trick is to say assume there are two solutions, $u$ and $v$. You then define $w=u-v$ and prove $w=0$, this shows that $u=v$ so the solutions are actually the same. If an expert is trying to show a solution to some poorly researched PDE is unique, you can bet that he/she will start with this strategy.
This is just one of many "tricks." This trick is so common that most people studying PDEs, even at only the advanced undergraduate level, have likely been exposed to this trick, and if not, can probably deduce this is a common way of proving a result.
$(3)$. Ingenuity/creativity
This cannot be taught. This is required when trying to prove something where the first two techniques are not working. For the Millenium Prize problems, not only do you have to likely chain dozens of statements together, each link in the chain is difficult to prove even to experts, and no one has an idea how to link them together.
