  PG 1553+113 is a very-high-energy (VHE, $E>100\,\mathrm{GeV}$) $\gamma$-ray
emitter classified as a BL Lac object. Its redshift is constrained by
intergalactic absorption lines in the range $0.4<z<0.58$. The MAGIC telescopes
have monitored the source's activity since 2005. In early 2012, PG 1553+113 was
found in a high-state, and later, in April of the same year, the source reached
its highest VHE flux state detected so far. Simultaneous observations carried
out in X-rays during 2012 April show similar flaring behaviour. In contrast,
the $\gamma$-ray flux at $E<100\,\mathrm{GeV}$ observed by Fermi-LAT is
compatible with steady emission. In this paper, a detailed study of the flaring
state is presented. The VHE spectrum shows clear curvature, being well fitted
either by a power law with an exponential cut-off or by a log-parabola. A
simple power-law fit hypothesis for the observed shape of the PG 1553+113 VHE
$\gamma$-ray spectrum is rejected with a high significance (fit probability
P=2.6 $\times 10^{-6}$). The observed curvature is compatible with the
extragalactic background light (EBL) imprint predicted by current generation
EBL models assuming a redshift $z\sim0.4$. New constraints on the redshift are
derived from the VHE spectrum. These constraints are compatible with previous
limits and suggest that the source is most likely located around the optical
lower limit, $z=0.4$, based on the detection of Ly$\alpha$ absorption. Finally,
we find that the synchrotron self-Compton (SSC) model gives a satisfactory
description of the observed multi-wavelength spectral energy distribution
during the flare.
