Why does the LP Formulation of the MST Problem need Topology Constraints?

I am looking for an example that demonstrates the necessity of either subtour-elimination or of connectivity constraints in the LP formulation of the MST

In the internet I only could find the LP formulations, but no motivation for the constraints.

The need for topological constraints seems contradictory to me, because demanding the number of edges to be $n-1$, together with the requirement, that at least one edge be adjacent to each vertex, suffice for a connected graph of $n$ vertices to ensure connectivity and a tree topology. Fractional solutions seem also counter-intuitive.

So, where does the need for topoligical constraints in the LP formulation of the MST problem actually come from, resp. when does it arise?

I'm not sure that I understand your question properly but the constraints you propose do not ensure connectivity. As far as I guess, you propose to use the following polytope to describe the spanning trees of a graph $G(V,E): \sum_{e\in E} x_e=|V|-1; \sum_{v\in e\in E} x_e\geq 1$ for all $v\in V$ and $x_e\geq 0, e\in E$. But then, for instance, in $K_5$ a triangle and a disjoint edge constitute a vertex of your polytope.