I suppose I thought it would need something to tell it to look at the contents of @A instead of @A itself.

Quite the reverse. You have to work at it in order to refer to the array rather than its contents. Consider these two separate lines from the code above:

print "intesection is @intersection\n";
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Here, despite being contained within a string, it is the contents of the array which are printed, not the array as a container.

my @A = (\@S1, \@S2, \@S3);
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Whereas here, we have had to create references to each of the S arrays so that they are not flattened when assigning to @A.

Further reading: Arrays: A Tutorial/Reference, perldata, perlref