Let's look at the first link in the chain, which is straightforward, but still deserving its clear treatment.

Each point in the manifold is called a phase, or state. So the manifold _is_ the state space.

In this whole discussion, we are assuming a deterministic system. That means that the "direction of motion" of the system is uniquely determined by the state. It's not random, and it doesn't depend upon time - just the state.

In a manifold, the notion of direction of motion at a point is expressed by a tangent vector. The space of all tangent vectors at a point is the tangent space at that point. It is a vector space.

Here we have a manifold \$$M\$$ consisting of states. For a given state \$$s \in M\$$, the tangent space \$$\mathit{Tangent}(s)\$$ consists of all the vectors for the possible directions of motion out of that state.

That the system is deterministic means that for each \$$s\$$, there is one uniquely determined vector in \$$M(s)\$$ that gives its direction of motion. Let \$$\Gamma(s) \in \mathit{Tangent}(s)\$$ be this uniquely determined vector.

\$$\Gamma(s)\$$ is a field of tangent vectors on the manifold.

This tangent vector field is a function that completely specifies the law of motion for the system.