Understanding least-fixed point logic
If you are having trouble with the concept of least fixed point, I would recommend spending some time getting a background in more general order theory.

Davey and Priestley, Introduction to Lattices and Order is a good intro.

To see why the transitive closure is the least fixed point, imagine building up the closure from an empty set, applying the logical formula one step at a time.  The least fixed point arrives when you can't add any new edges using the formula.

The requirement that the formula be positive ensures that the process is monotonic, i.e. that it grows at each step.  If you had a negative subformula, you could have the case where on some steps the set of edges would decrease, and this could lead to a non-terminating oscillation up and down, rather than a convergence to the LFP.