I think the intuition is that we want to focus on the edges of the box (where the lengths sum together like the answer we're looking for), not the volume (where the lengths multiply together and cause problems). So we expand the edges until they completely dominate the equation.
0 < (Vol(Bε) - Vol(Aε))/ε² = ((a+b+c) - (a'+b'+c'))π + 2((ab + ac + bc) - (a'b' + a'c' + b'c'))/ε + (abc - a'b'c')/ε²
The limit for ε to infinity must be >= 0 and is ((a+b+c) - (a'+b'+c'))π, therefore a+b+c >= a'+b'+c'.
"large epsilon" is a standard way to express "consider the asymptotic behavior for epsilon to infinity and you will see what I mean".