I'm getting my feet wet with Seifert Fibered Spaces in Hatcher's 3-manifold papers. Elsewhere, it is said that this follows easily from the definition. I am not seeing it. I think we would need to know that $T^2$ and $K^2$ are the only surfaces which have a foliation of circles.
Both surfaces can be described foliating over a (the yellow) circle as a base (the Zerlegung) by circles (in reds and blues). In the picture the "leaves" (in reds and blues) meet the base in yellow. The edges of the squares are identified regarding the specified letters and the directions of them the get $T$ and $K$ respectively.