The P vs. NP problem is a question about Turing machines $T$, because the complexity classes P and NP are defined in terms of these theoretical machines. Let's call these classes $P_T$ and $NP_T$ from now on. The paper introduces a new theoretical computation machine $H$ that has associated classes $P_H$ (runs in polynomial time on the hyperbolic cellular automaton) and $NP_H$ (runs in non-deterministic polynomial time on the hyperbolic cellular automaton).
The first step in this paper is a proof that the 3SAT problem, a well known $NP_T$-complete problem, can be solved in polynomial time on this machine, i.e. this problem lies in $P_H$. Next, they show any polynomial time reduction on a Turing machine can be performed in polynomial time on their hyperbolic automaton. Since 3SAT is $NP_T$-complete, any $NP_T$ instance can be reduced to a 3SAT instance in polynomial time (on $T$ by definition, so also on $H$ by their lemma) and then be solved by solving 3SAT in poly-time, both on the hyperbolic automaton. In other words, the main result of this paper (Theorem 1) can be written as $NP_T \subseteq P_H$ in our notation. This does not give a solution to the P vs. NP problem, because that would need to relate the classes $NP_T$ and $P_T$.
Note that the authors include some remarks on the P vs. NP problem in section 4.2, where they claim their result is evidence for P$\neq$NP (!):
A third direction consists of the new light shed on the P=NP question in the
ordinary settings. As the hyperbolic space has properties which are very different from the properties of the euclidean space, in particular, it has many more directions, would not that be a hint favorable to prove that P$\neq$NP in euclidean conditions? It seems that
for the last ten years the works in the field of complexity incline people to believe
more in P$\neq$NP. Apparently, the present result also belongs to that trend.