Consider LTI causal and finite-dimensional system $G$ and let $\hat{G}$ be its Laplace transform.
We say that $\hat{G}$ is stable if it is analytic in the closed right half-plane (Re $s>0$).
What "analytic" means here?
I know that $\hat{G}$ is stable if doesn't have poles in RHP. Also I found wiki definition for analytic function. But I couldn't relate them.
The Wikipedia definition is exactly right.
Your system is a linear time-invariant system of differential equations in finitely many variables. The Laplace transform of such a system is a rational function, and thus is analytic except for poles. It is stable if none of those poles are in the closed right half plane (which, BTW, is ${\text Re}(s) \ge 0$, not $>0$).