Friendly reminder that from a math perspective probabilities in logistic regression are almost exactly the same as velocities in special relativity. If you understand log odds ratios then you secretly know the basics of special relativity! That may be why the former is so hard... 
Another way of thinking about it -- logistic probabilities add in almost exactly the same weird way that relativistic velocities do. But everything adds _approximately_ linearly around p = 1/2 and v/c = 0. Keep that in mind the next time you use a linear probability model...
For the non-physicists who want to impress their friends: the axes x^- and x^+ in the right plot are light-cone coordinates and the straight lines are the trajectories of objects moving at constant velocities. w is the rapidity, an unconstrained relativistic velocity.
For the physicists who want to learn statistics: the left plot shows the relationship between probabilities, odds, and log odds ratios (or logits) that arise when putting a solid statistical foundation on logistic regression models.
