Thank you for your question. It motivated me to go through that
paper. While I don't have a direct answer, I broke the paper
down. The difficult details are non-omitable.. you have to find
them in the paper.

---

Dvoretzky-Kiefer-Wolfowitz inequality is really about bounding
the $\infty$-norm (sup-norm) of the Kolmogorov distribution
$D_{n} := sup_{x\in R} |\hat{F}_{n}(x) - F(x)|$, where $F$ is the
cdf and $\hat{F}_{n}$ is the empirical distribution function) of
a single variate distribution.

To my surprise, Kolmogorov distribution is distribution-free,
meaning that it does not depend on the underlying cdf! And this
is actually not difficult to see - Hints:
1. Any random variable
$X$ with continuous cdf $F$ satisfies that $F(X)$ is a
$Unif_{[0,1]}$!
2. Replace $x$ with $F^{-1}(t)$ and $x \in
\mathbb{R}$ with $t \in [0,1]$ in the definition $D_{n}$. (This
argument should also work for other $L_{p}$-norms, but I still
can't find a version of DKL inequality for them.)

Therefore, in this paper, the author first assumed that $F$ is
the cdf of the uniform disribution on [0,1]. Then the bounding
problem becomes very explicit. And that's why it is about the
Kolmogorov-Smirnoff distribution. It is very natural to inspect
this due to the reduction above.

### Main Steps

We follow the notations in the paper. $D_{n}$ is easily
transformed to $D_{n}^{-}$, which the paper will focus. The main
theorem (DKW with sharp bound) is

+ Theorem 1: $P(D_{n}^{-} > \lambda) \leq exp(-2 \lambda^{2})$,
  under some very mild conditions.

The key idea is to compare $D_{n}^{-}$ with some quantities
$f_{\lambda}(s)$ derived from Brownian bridge.

More precisely, we have
+ (2.3) $P(D_{n}^{-} > \lambda) = \sum_{0 \leq j \leq n - \lambda \sqrt{n}} p_{\lambda,n}(j)$
+ (2.4) $exp(-2\lambda^{2}) = \int_{0}^{1} f_{\lambda}(s) ds$
+ (Prop. 1) compares $p_{\lambda,n}(j) and f_{\lambda}(s)$

Finally, in the last 5 pages, the author broke things down to two
cases and proved them separately by hard estimation.

1. $n \geq 39$ and $\lambda \leq \sqrt(n)/2$
2. $n \leq 38$ or  $\lambda > \sqrt(n)/2$