Zeroth homotopy group: what exactly is it?

This is the definition.

So $\pi_0$ is the homotopy classes of maps from two points ($S^0$) to $X$, where the first point is mapped to the base point. Clearly only the path connected component matters for the second point (since a path connecting the two points defines a homotopy between two such maps).

$\pi_0$ being trivial implies that there is a path between any point and the base point, i.e. $X$ is path-connected.

For a space $X$ and a base point $x_0$ , define $\pi_n(X,x_0)$ to be the homotopy class of maps $f:(I^n, \partial I^n) \to (X,x_0)$ , where homotopies $f_t$ are required to satisfy $f_t(\partial I^n)= x_0$ . this definition can be extended to the case n=0 by taking $I^0$ to be a point and its boundary to be an empty set, so $\pi_0(X,x_0)$ is just the set of path connected component.

Just a slight rephrase: you can consider $\pi_0(X)$ as the quotient set of the set of all points in $X$ where you mod out by the equivalence relation that identifies two points if there is a path between them.