  We study the Berkovich analytification of the space of genus $0$ logarithmic
stable maps to a toric variety $X$ and present applications to both algebraic
and tropical geometry. On algebraic side, insights from tropical geometry give
two new geometric descriptions of this space of maps -- (1) as an explicit
toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical
compactification in a toric variety. On the combinatorial side, we prove that
the tropicalization of the space of genus $0$ logarithmic stable maps coincides
with the space of tropical stable maps, giving a large new collection of
examples of faithful tropicalizations for moduli. Moreover, we identify the
optimal settings in which the tropicalization of the moduli space of maps is
faithful. The Nishinou--Siebert correspondence theorem is shown to be a
consequence of this geometric connection between the algebraic and tropical
moduli spaces.
