Meaning of the Soul Theorem The Soul Theorem states that in every complete, connected riemannian manifold $M$ with $\mathrm{sec}(M)\geq 0$, there exists compact, totally convex, totally geodesic submanifold $S$ such that $M$ is diffeomorphic to the normal bundle of $S$. 
I don't know if I'm being too formal on my approach but I don't totally get how a manifold could be diffeomorphic to a bundle given that they're in different classes of objects. What is the precise meaning of diffeomorphic here? In general the normal bundle of $S$ (any submanifold) will be of dimension les than $\mathrm{dim}(M)$ right? So how is it possible that they're diffeomorphic?

The total space of any vector bundle over a manifold is itself a manifold; if it has rank $r$ and the base space has dimension $n$, it has dimension $r+n$ (since it locally looks like a product). Since the rank of the normal bundle $NS$ is equal to the codimension of $S$ in $M$, the dimension of $NS$ as a manifold is equal to the dimension of $M$.
The general version of the picture you want to be visualizing (which is more elementary than the Soul Theorem) is the tubular neighborhood theorem; one phrasing of it states that  if $A$ is a submanifold of $M$ then $A$ has a neighborhood in $M$ which is diffeomorphic to $NA$.
