Plain English Meaning
If the next part of your route requires a higher altitude for terrain or obstacle clearance, the minimum crossing altitude is the altitude you must already be at by the time you reach the fix where that next segment begins. You can't wait until after the fix to climb -- you have to be there by then.
Full Definition
The lowest altitude at certain fixes at which an aircraft must cross when proceeding in the direction of a higher minimum en route altitude (MEA). It ensures that an aircraft climbing toward a higher MEA segment has reached an altitude that provides required obstacle clearance before crossing the fix into that next segment.
Context Anchor
Seen on instrument flight rules en route charts and procedures, usually at a named where the route ahead requires a higher altitude.