Please could somebody put the following line into English.

next if $seen{ $elem }++;
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My attempt is:
Insert $elem into the hash 'seen' and skip to the next loop event if this $elem is already present in the hash 'seen'.

How 'bout exectly the way you read it, i.e. "skip to the next iteration of the loop if you have already seen $elem"? That is "if the counter associated to $elem (that is the value of the hash %seen on the key $elem), which you're (post-)incrementing (++) at this iteration too, is greater than zero".

Was there anything particular difficult to understand in this reply?