Let $\Phi$ be a root system of a Euclidean space $E$. The Weyl Group is the subgroup of $GL(E)$ generated by relections $\sigma_\alpha$, $\alpha\in\Phi$.
By definition, a reflection is an element of $GL(E)$ fixing pointwise a hyperplane $$P_\alpha=\{w\in E: (w,\alpha)=0 \}$$ and sending $\alpha$ to $-\alpha$.
But the identity element of a subgroup is the identity element of the full group. But the identity of $GL(E)$ is obviously not a reflection; in particular it does not send any $u\in E$ to $-u$.
How, then is the Weyl Group a group?
If $G$ is a group and $S\subseteq G$ a subset, then the subgroup generated by $S$ is the subgroup which is minimal among all subgroups containing $S$. There is a unique minimal one, it is just the intersection of all subgroups containing $S$, which there is at least one of ($G$ itself). The notation for it is $\langle S\rangle$, and its elements are all elements of $G$ that can be formed using products of elements from $S$ and their inverses (we need to include inverses in general since otherwise we may only generate a submonoid without all the inverses it needs). This includes the "empty product" of zero-many elements, which will by definition by the identity element of $G$.
Thus, even if $S$ is not itself a subgroup (like any set of reflections), $\langle S\rangle$ is.
In the case of reflections, they are self-inverse so we don't need to worry about specifying inverses, and that also means the identity is the product of any reflection with itself (if you don't like the idea of an empty product). In general, the product of two distinct reflections will be a rotation, which is not a reflection. It is a good exercise to explicitly describe the rotation which results from the composition of two reflections using geometry. This can lead to the Cartan-Dieudonne theorem.
For a simpler version of the exercise, consider geometrically describing the composition of two reflections in a finite dihedral group. It will be a rotation, but by what angle? By slicing space up appropriately, this can be generalized to the $n$-dimensional situation.