I don't understand how the zeros are found of this 2nd order feed-forward filter
Below is a polynomial in $z$ :
$$ P(z) = 1+\alpha_1z^{-1}+\alpha_2z^{-2} $$
Zeros of this polynomial is also known as its roots; i.e., those $z_r$ for which $P(z_r)=0$.
Solving for the roots of a general $n$-th order polynomial is not possible. They are numerically found.
However, one alternative way to define a polynomial is to write it as the product of factors of the form $1 - d_k z^{-1}$, where each $d_k$ is a zero/root of $P(z)$; i.e., $(1 - d_k d_k^{-1}) = 1-1 = 0$.
Then your second order polynomial $P(z)$ can be written as
$P(z) = (1-Z_1z^{-1})(1-Z_2z^{-1})$
where the complex valued $Z_1=Re^{j\theta}$ and $Z_2=Re^{-j\theta}$ are the roots of the polynomial. You should solve the second order equation $P(z)=0$ to find out what those zeros are.