I believe the image below is sufficient for understanding the problem but if you need more context, everything can be found at page 42 of https://www.math.ucla.edu/~yanovsky/handbooks/PDEs.pdf
I need to know why the absolute value of each integral is less than the absolute value of the maximum in the integration set. Is this a theorem from calculus?
I first thought about the maximum principle for harmonic functions but I don't think $u$ is harmonic and also I don't see how it involves integrals.
The title of your post is not correct (and not relevant to the body of your question).
I think the result you are looking for is $$\int_A |f(x)| \, dx \le \max_{z \in A} |f(z)| \cdot \int_A 1\, dx$$ which comes from bounding $|f(x)| \le \max_{z \in A} |f(z)|$ for any $x \in A$.
The integrals in your question are surface integrals, so the bound is the maximum of the integrand on that set, times the volume/area of the surface (that is the $\omega_n \epsilon^{n-1}$ term).