The following theorem is proved in Milnor's famous book "Morse theory".
Theorem 21.7 (Bott). Let $G$ be a compact, simply connected Lie group. Then the loop space of $G$ has the homotopy type of a CW-complex with no odd dimensional cells.
It is not clear to me where the author uses the simply connectedness of $G$. Is it a necessary condition? Can someone please illuminate?
You can easily drop the connectivity condition as long as each (equivalently, one) component is simply-connected. But the assumption that $\pi_1(X)=1$, is a necessary condition. If $X$ is a complex without odd-dimensional cells then $\pi_1(X)=1$: Indeed, by the cellular approximation theorem, every loop $c$ in $X$ is homotopic to a loop $c'$ in $X^1$. If $X^1$ contains no 1-cells, then $X^1=X^0$, implying that $c'$ is constant. Thus, $\pi_1(X)=1$.