Horizontal thinking discussed here is in terms of managing projects as popularized by David Allen of David Allen Co.
In horizontal thinking, you must look across all the hundreds of parts of dozens of things you need to keep tabs on during the course of any 24-hour period. It often requires that you be extremely flexible in recalibrating when to do "what actions" on multiple things you have going at any given time. The requirements and actions for horizontal usually blows the heck out of our vertical thinking, and is usually considered the "not-so-fun" part of doing work and executing necessary tasks to make the horizontal vision a reality.
One way David Allen describes horizontal thinking is like a police scanner- the kind that scan back and forth looking for a signal. This is how your brain operates when dealing with the reality of horizontal day-to-day activities.
See also vertical thinking.