A cobordism between two manifolds represents some kind of evolution from one state space to another state space. In quantum mechanics we have an evolution in a continuous time parameter. In order to get a TQFT that describes this, choose

0-dim topological manifolds, i.e. points, as objects (or "the point" as the only object)

1-dim cobordisms of the points as arrows,

add a Riemannian structure to the 1-dim cobordisms, which equips them with a length.

Now a TQFT on this category will be a functor that

associates a (finite or infinite dimensional) vector space $H$ to the point,

associates a linear operator $U(t): H \to H$ to every (or the) interval of length $t$ such that

$U(t s) = U(t) U(s)$ (functoriality).

For infinite dimensional vector spaces we would need to assume that they are (complex) Hilbert spaces. And we would in addition have to assume that the linear operators that are associated to cobordisms are unitary operators. Then the TQFT is identical to the Schrödinger picture of quantum mechanics. Rays in the vector space $H$ represent the state of a physical system and the operators $U(t)$ represent the evolution of the system from one state at the time 0 to the time $t$.

Edit: In the physical interpretation of TQFT the vector spaces represent **state spaces** of the physical systems which is not the same as physical space. A point in a state space represents all information that is necessary to completely describe the physical system at hand.