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I understand how this transfer function was solved except I don't know how to get the 0.87. Any help would be appreciated.

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In short: $(0.937)^2=0.878$

A bit more detail:

$$\begin{align}(z-re^{j\theta})(z-re^{-j\theta})&=z^2-r\left(e^{j\theta}+e^{-j\theta}\right)z+r^2\\&=z^2-2r\cos(\theta)z+r^2\end{align}$$

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