The point is that we'd like to define a function "sorta like square root" with the following property:
If $\gamma: [a, b] \to \mathbb C - \{0\}$ is a curve, then
$F(\gamma(t))^2 = \gamma(t)$ (so $F(z)$ is a square root of $z$), and
$t \mapsto F(\gamma(t))$ is continuous as a function of $t$.
In other words, the proposed function $F$ would let us compute square roots "in a continuous way depending on position."
Alas, that's not possible for any function $F: \mathbb C \to \mathbb C$, for if $\gamma$ traverses the unit circle once, we find that $F(1)$ must be both $+1$ and $-1$.
On the other hand, we can build a surface $M$ with the property that
There's a function $p: M \to \mathbb C$ that's smooth and is 2-to-1 almost everywhere (i.e., for each point of $\mathbb C$, there are two points of $M$), the exception being that $p^{-1}(0)$ consists of a single point of $M$.
every continuous path in $\mathbb C - \{0]\}$ has exactly two "lifts" to $M$, i.e., for a such a path $\gamma$, there's a continuous path $\alpha: [a,b] \to M$ with $p(\alpha(t)) = \gamma(t)$ for every $t$. (And in fact there are two of these).
And on this "Riemann surface" $M$, there is a function like the one we were looking for, i.e., there's a continuous function
$$
F: M \to \mathbb C
$$
with the property that if $\alpha$ is a lift of a curve $\gamma$ that misses the origin, then
$$
F(\alpha(t))^2 = \gamma(t)
$$
for every $t$. So on this Riemann surface, "there are square roots".
