-a Links two expressions together in an “and” or “&&” expression. This option is deprecated.
-n Checks if the length of a string is nonzero.
You could translate the test expression into the following pseudocode:
if ( ($1 has nonzero length) and
($2 has nonzero length) and
($3 has nonzero length) )
There are no checks in that expression for whether the file exists or doesn’t exist, only whether the arguments have been supplied to the script.
The arguments -a and -n can be found in the manpage for test
man test
The operator [ ... ] is often used as shorthand for test ... and likely has identical functionality on your system.