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Can someone please explain what happened here? I know what happens, but I don't get this math


I believe we are assuming that $\vec{n}$ is a unit vector here. That being said, note that $(\omega_{i}\cdot \vec{n})\vec{n}$ is the projection of $\omega_{i}$ onto $\vec{n}$, so $2(\omega_{i}\cdot \vec{n})\vec{n}$ is basically sticking $2\omega_{i}$ in the direction of the normal. Because $\theta_{i} = \theta_{o}$, and we're assuming a perfect specular reflection, this is the same thing as the vector $\omega_{i} + \omega_{o}$.

$\omega_{o} = -\omega_{i} + 2(\omega_{i}\cdot \vec{n})\vec{n}$ just follows from moving $\omega_{i}$ to the RHS of the equation.