Unfortunately this is not true. For instance $\mathbb{R}^n$ is simply connected and, being an infinite abelian group, is very far from being simple. When $n > 1$ it is not even simple as a Lie group, i.e., it has nontrivial connected, closed normal subgroups.
Also the other direction is false: $\operatorname{PSL}_2(\mathbb{R})$ is simple as an abstract group, but not simply connected, since $\operatorname{SL}_2(\mathbb{R}) \rightarrow \operatorname{PSL}_2(\mathbb{R})$ is a degree $2$ connected covering map. (If am not mistaken, then at least for linear groups what is true is that an abstractly simple group is of adjoint type, i.e., the dual condition to being simply connected.)
Anyway, you did the right thing: if you have a healthy level of interactions with other members of the mathematical community (students, faculty, etc.) then you will "hear things" which are quite new to you. Not everything you hear is actually true -- or especially, what you heard / remembered is not actually true -- so it's important to try to verify / disprove these things that you hear, or ask someone else about them.