Why should I track lie types for putts?

This allows you to record more detailed putting and Strokes Gained statistics.

How do I select the lie type for a putt?

Make your selection on the same screen you would use to set your club. 

Next, select Putter and choose a lie type from the top of the screen. The default lie for a putt is "On green", if you used your putter from off the green simply select the correct lie type from the menu.