Roots in modulo field
You want to find a point $(X,Y)$ on an elliptic curve $y^2 = x^3 + ax + b$ knowing only $X$ and a single bit indicating whether $Y$ is even or odd. To find $Y$, you use the relation defining the curve: you know that $Y^2 = X^3 + aX + b$ since the point is on the curve. So you compute $X^3 + aX + b$ using your value of $X$ and the public parameters $a, b$, and you want to find values of $Y$ such that $Y^2$ is equal to that value.

If the modulus $p$ is congruent to $3$ mod $4$, that's very easy to do: for any $z$, the square roots of $z$, if they exist, are $\pm z^{(p+1)/4} \bmod{p}$. Indeed, suppose $z$ has a square root, say $a$. Let $g$ be a primitive root mod $p$, and let $a = g^x$. Then $z = (g^x)^2 = g^{2x}$. In turn,

$$(z^{(p+1)/4})^2 = z^{(p+1)/2} = (g^{2x})^{(p+1)/2} = (g^x)^{p+1} = (g^x)^{p-1}g^{2x} = g^{2x} = z$$

which shows that $z^{(p+1)/4}$ is indeed a square root of $z$.