Question about theorem on homotopic curves for a holoorphic function. (stein and shakarchi)

I don't have the book, but I assume the idea of the proof will be something similar to the following: Look at the picture you provided. It's a drawing of the image of $F(s,t)$, where the lines drawn correspond to constant $s$. You could now add more lines corresponding to constant $t$ to get something looking like a net covering the image of $F$. The idea is to show that you can make the net fine enough to allow each "hole" in the net to be contained in a disc of radius $varepsilon$, which in turn is contained in $Omega$. Then you can apply Cauchy's integral theorem to the integral along the boundary of a single hole, and then add the integrals along the boundaries of every hole to get the integral along the boundary of the entire net, which will then be $0$, so the integrals along $gamma_0$ and $gamma_1$ will be the same.

To actually use this idea, though, they first need to establish that there exists an $varepsilon>0$ small enough that such discs centered at points in the image of $F$ actually are contained in $Omega$. That's what they're doing. Using $3varepsilon$ instead of $varepsilon$ is probably so that they can use a triangle inequality in some way to get discs of radius $varepsilon$ in the end. But right now, they're trying to show that there is some radius ($3varepsilon$) such that every disc with that radius whose center lies in $K$ is contained in $Omega$. Essentially, there is a minimum distance between the boundaries of $K$ and $Omega$ greater than $0$.

They are proving that, for some $varepsilon>0$, every disk with radius $3varepsilon$ centered at a point of $Fbigl([0,1]times[a,b]bigr)$ is a subset of $Omega$.