My thoughts on proving this statement is as follows:
Suppose $G_a$ is an open cover of $Q= [0,1] \times [0,1]$. For each $x$ in $[0,1]$, there is some ball around $x$ with radius $r_x$ such that it covers $[x-r_x, x+r_x] \times [0,1]$. Since $[0,1]$ is closed and bounded, it is compact. Since $[0,1]$ is compact, this vertical strip described above is also compact, i.e., there is a finite collection of these balls of $G_a$ that covers this vertical strip.
Next we do the same thing except for horizontal strips.
This is where I am drawing a blank, how am I to show that the union of these balls are finite, and how would I know that it covers all of the unit square?