With Monte Carlo off-policy learning what do we correct by using importance sampling? I do not understand the link of importance sampling to Monte Carlo off-policy learning.
We estimate a value using sampling on whole episodes, and we take these values to construct the target policy.
So, it is possible that in the target policy, we could have state values (or state action values) coming from different trajectories.
If the above is true, and if the values depend on the subsequent actions (the behavior policy), there is something wrong there, or else, better, something I do not understand.
Linking this question with importance sampling, do we use this ro value to correct this inconsistency?
Any clarification is welcome.
We estimate a value using sampling on whole episodes, and we take this values to construct the target policy.

The crucial bit that you are missing is that there is no single value of $V(s)$ (or $Q(s,a)$) of a state (or a state action pair). These value functions are always defined with respect to some policy $\pi(a|s)$ and is given the notation of $V^{\pi}(s)$ (or $Q^{\pi}(s,a)$).
The off-policy learning problems are arising when you have two policies: the generation policy $\mu(a|s)$ and the target policy $\pi(a|s)$. Your MC sampling data came from an agent following $\mu$, while you want to improve your target policy $\pi$. It is pretty straightforward from here that you'd need to weight your calculations with factors like $\frac{\pi(a_i|s_i)}{\mu(a_i|s_i)}$ - that's what importance sampling is.