Why is a reflection followed by another reflection is a rotation? I just started abstract algebra and we are working with dihedral groups. I've made Cayley tables for D3 and D4 but I can't explain why two reflections are the same as a rotation

It all depends on what you mean by "reflection/rotation."
Quite often you say that a rotation is an orthogonal transformation with determinant $1$, and a reflection is an orthogonal transformation with determinant $-1$. By multiplicatively of determinant, this explains why the product of two reflections is a rotation. This works if you consider your dihedral group as a subgroup of linear transformations on $\mathbb R^2$.
Just thinking in terms of the structure of the dihedral group, the fact that the subgroup of rotations has index $2$ explains why the product of any two reflections (in the sense of a dihedral group) is a rotation. If $R$ is the rotation subgroup and $x,y$ are reflections, then $xR=yR$ and $xR•xR=R$ imply $xR•yR=xyR=R$, that is, $xy\in R$.
