Why the Euler characteristics of a compact connected lagrangian submanifold of $\mathbb{R}^4$ is zero?

Question

Let's consider space $\mathbb{R}^4$ with the standard symplectic structure. There is a fact that if $L\subseteq \mathbb{R}^4$ is an embedded compact connected lagrangian submanifold, then it is a torus, in other words, $\chi(L)=0$. I saw this fact in many books and articles, but all proofs I could find are either not very detailed or too difficult for me. Could you write this proof using the simplest methods.
I will give some information about my knowledge and problems that I have reading proofs of this fact.
I know about Weinstein's neighborhood theorem. I understood that I have to use that some neighborhood of $L$ is symplectomorphic to some neighborhood of zero-section in cotangent bundle $T^*L$. Also, many proofs use isomorphism $TL \cong T^* L\cong N L$. I understand that these isomorphisms exist, but in proofs are used such phrases as "canonical" isomorphism and "orientation preserving/reversing" isomorphism, which I understand not confidently and do not understand if it is important or not. Then, I know about Euler characteristics as a number of zeros (with signed or modulo two) of a generic vector field on a manifold. Also, I am a bit familiar with the intersection index of two submanifolds, so as I understand Euler characteristics of submanifold is the intersection index of the zero-section in its tangent bundle with itself, but I am not sure about a sign in this formula. In some proofs are used homology classes, but I do not know this theory enough to use it.
In general, it seems to me after reading all proofs I found that I have all instruments to prove this fact, but some steps I do not see or understand.