print map "$_\n", keys %{ {map {$_ => 1} @array} };
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Alrighty, let's break it down then

map { $_ => 1 } @array
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For each value in @array apply the transform in the block (denoted by the curly braces), in this case create a two element list consisting of each element of the list and a literal 1, collect the output of those transforms and return them as a list. So the map creates a list of key-value pairs (although the value is superfluous in this case) perfect for a hash ...

{ map { $_ => 1 } @array }
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The extra curly braces around the map statement create an anonymous hash, so all duplicate keys in the list created by map are collapsed in the anonymous hash.

keys %{ {map {$_ => 1} @array} }
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Dereference the anonymous hash created and return a list of its keys.

print map "$_\n", keys %{ {map {$_ => 1} @array} };
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The final map then simply a creates a list where each key has a \n appended and then print prints that list.
HTH

_________
broquaint

Thank you for all the explanations!