Suppose we have an arbitrary graph with $l$ vertices. Suppose we have $n$ vertices with odd degree. Let's consider what happens when we connect(or remove) an edge between any two vertices, there're two possibilities:
1) The two vertices are both of even(or odd) degree: Suppose both vertices are even degree. Connecting(or removing) an edge between them will increase the degree of both vertices by $1$ (or decrease in case of removing an edge), therefore both vertices will become odd degree, and $n$ will increase by $2$.
Similarly, if the two vertices were initially odd degree, then connecting(or removing) an edge will turn both of the vertices into even degree, and $n$ will decrease by $2$.
2) One vertex is odd while the other is even. In this case, connecting(or removing) an edge will turn the odd vertex into an even one, and the odd vertex becomes even. Therefore $n$ will remain unchanged.
From this we learn that the number of odd degree vertices $n$ can only increase/decrease in steps of $2$(or remain unchanged) as we create(remove) edges.
Now when we construct any graph we wish, we start with our vertices isolated from each other with not a single connection(edge). Therefore, initially $n=0$(all vertices are of even degree since not a single edge connects them) , and as we make connections, it can only increase/decrease in steps of $2$, therefore $n$ will remain even.