Sudo (su "do") allows a system administrator to delegate authority to give certain users (or groups of users) the ability to run some (or all) commands as root or another user while providing an audit trail of the commands and their arguments.
I'm not sure that you are confused about this, but you might be based on the question.
If you're only confused about why
Is the same as
This is because in UNIX, the . means the current directory. So in this way, you will not search your PATH variable for the nearest mysqld_safe and run it, you will specifically run the one at /usr/local/mysql/bin/.