Cluster Authentication Question

I'm following this guide to setup an Ubuntu MPICH2 Cluster (https://help.ubuntu.com/community/MpichCluster). Under section 5 (5. Defining a user for running MPI programs) the guide says "We define a user with same name and same userid in all nodes with a home directory in /mirror." My question is why does the userid need to be the same? What is the process that linux uses to determine what users are authorized to make changes? What files are involved this process? Any help in understanding would be appreciated!