Why does the edge length information shown after ticking Edge Info -> Edge Length under Mesh Display change after choosing Global or Local?
When you are using Local, you are looking at the dimensions relative to the scaling of the object (if scaled in object mode).
Global will show you the absolute measurements.
As an example. Here's a cube that has been scaled 2 times in the $Z$ axis.
If you switch between local and global coordinates in edit mode.
The measurements in local units are relative to the scaling and show that the cube is 2x2x2, whereas the Global units show the absolute measurements showing that the cube is no longer a cube, but a cuboid that is 4m tall.
Why should you care for local or global measurements?
Global is based on the 'World' coordinates (0,0) where 'Local' is relative to its self, which could be anywhere in 'World' space. [hope i got that right].