Someone once suggested on MO that this is because on the one hand Matiyasevich's theorem shows that no algorithm can solve Diophantine equations over $\mathbb{Z}$ (and the corresponding result is not known over $\mathbb{Q}$), and on the other hand this is possible over $\mathbb{R}$ because of quantifier elimination. It is also possible over $\mathbb{Q}_p$ for every $p$, and this suggests that one would really like to have some sort of local-to-global principle so that special types of Diophantine equations can be solved, and if not, to understand how it fails...
Anyway, I am still not sure I totally agree with the sentiment of the first sentence. Think of it this way: if I want to understand a category $C$ and I understand a category $D$ very well, and moreover I have a functor $F : C \to D$, then it stands to reason that I can learn something about a problem in $C$ by translating it to a problem in $D$, where I understand what is going on very well. Nothing is mysterious about this process other than possibly the construction of the functor $F$, and in the case of number theory $F$ is something like the analytification functor from varieties over $\mathbb{Z}$ or $\mathbb{Q}$ to varieties over $\mathbb{R}$ or $\mathbb{C}$ and the existence of this functor is not so hard to understand.
Someone once suggested on MO that this is because on the one hand Matiyasevich's theorem shows that no algorithm can solve Diophantine equations over $\mathbb{Z}$ (and the corresponding result is not known over $\mathbb{Q}$), and on the other hand this is possible over $\mathbb{R}$ because of quantifier elimination. It is also possible over $\mathbb{Q}_p$ for every $p$, and this suggests that one would really like to have some sort of local-to-global principle so that special types of Diophantine equations can be solved, and if not, to understand how it fails...