I borrow this proof from [Birkenhake-Lange, Complex Abelian Varieties, Lemma 1.1.1].
Let $X$ be a projective variety having a group structure. I assume that we are working over $\mathbb{C}$.
Consider the commutator map $\Phi(x,y)=xyx^{-1}y^{-1}$, and let $U$ be a coordinate neighborhood of $1 \in X$. By the continuity of $\Phi$, and since $\Phi(x,1) =1 \in U$, for all $x \in X$ we can find open neighborhoods $U_x$ and $W_x$ such that $\Phi(U_x, W_x) \subset U$.
Since $X$ is compact, finitely many $V_x$ cover $X$. Calling $W$ the intersection of the corresponding subsets $W_x$, we get $\Phi(X, W) \subset U$.
Now $\Phi(1, y)=1$ for all $y \in W$. Since holomorphic functions on a compact variety are constant, it follows $\Phi(X, W)\equiv 1$. Being $W$ open and non-empty, this in turn implies $\Phi(X, X) \equiv 1$, which is our claim.
Notice that "projective" is not really necessary, in fact what we actually use in the proof is "compact complex". Indeed, pushing further this argument (by a straightforward use of the exponential map) one can show that any compact complex connected Lie group is a complex torus.