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.
