In a nutshell..
/var is where logs are stored.
/tmp is able to be written to by anyone.
/home is where user data is stored.
/usr is typically where software is installed.
If these locations were stored on a single filesystem, that filesystem could be fully utilized. If
/ were full, it could potentially prevent the system from operating properly.
Additionally, separate filesystems can be mounted with different settings. For example, I like to mount
nosuid. If an application were compromised and had files written to
/tmp, not being able to execute files would distinctly limit the ability to further compromise the system.
I would recommend reading more about the Filesystem Hierarchy Standard to better understand those locations, as my description is summarized and not complete.