It looks to me like a discrete version of Gronwall's inequality.
If you have a sequence of numbers satisfying
$$ E_n \leq k \Delta t + (1 + \ell \Delta t) E_{n-1} $$
You can rewrite
$$ A_n = (1 + \ell \Delta t)^{-n} E_n $$
to get
$$ A_n \leq \frac{k \Delta t}{(1 + \ell \Delta t)^n} + A_{n-1} $$
which implies
$$ A_n \leq A_0 + k\Delta t \sum_{m = 1}^n \frac{1}{(1 + \ell \Delta t)^m} $$
The sum is a geometric series bounded by $\dfrac{1}{\ell \Delta t ( 1 + \ell \Delta t)} \leq \dfrac{1}{\ell \Delta t}$$\dfrac{1}{\ell \Delta t}$ (see comment below for the computation)
So you conclude
$$ A_n \leq A_0 + \frac{k}{\ell} $$
This gives
$$ E_n \leq (1 + \ell \Delta t)^n ( E_0 + \frac{k}{\ell} ) $$
which is exactly what is claimed, using that in the above derivation you can use
$$ k = C_3 \max \| f^{n+1} \|^2 $$
and
$$ \ell = C_1 + C_2 $$