How to prove this equality with rotation matrix

Since you seem to know that any rotation is orthogonal, it is pretty easy: An orthogonal matrix satisfies $R^TR=I$ or equivalently $R^T = R^{-1}$ which is either your definition of an orthogonal matrix or a consequence of your definition (it might be the case that you defined orthogonal matrices in a different way).

Then we can calculate $$||v||^2 = \langle v, v \rangle = \langle v, R^TRv \rangle = \langle Rv, Rv \rangle = ||Rv||^2$$ and by taking the square root we have what we wanted to show.

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