What is the geometric interpretation of this limit :$\lim_{n\to {+\infty}}\zeta(n)+\zeta(\frac1n)=\frac12$?

There is no deep underlying geometric picture. The fact that $\zeta(n) \to 1$ as $n \to \infty$ is nothing more than the fact that the first term is $1$, and every other term decays. Similarly, $\zeta(x)$ is continuous at $x = 0$, and $\zeta(0) = -1/2$. For two quite simple reasons, we see that $\zeta(n) + \zeta(1/n) \to 1 - \frac{1}{2} = \frac{1}{2}$.

It may be beneficial to ruin some of the artificial symmetry. It is also true that $$ \lim_{n \to \infty} \zeta(3n) + \zeta(\tfrac{1}{7n}) = \frac{1}{2}.$$ Or instead of $3$ and $7$, you can use any positive real numbers you want. I think the complete freedom in this choice shows how unrelated $\zeta(n)$ and $\zeta(1/m)$ really are.