map +{ NAME => $_ }, LIST
is another way of writing
map { { NAME => $_ } } LIST

The expression { NAME => $_ }

1. creates a hash,
2. assigns NAME => $_ to it (creating a key named NAME whose associated value is $_), and
3. returns a reference to the hash.

The expression map { { NAME => $_ } } @files

1. For every element of @files,
   1. sets $_ to the element
   2. executes the sub-expression.
2. A list is formed from the result of the sub-expressions and returned.

In other words,

$template->param(
   FILES => [ map +{ NAME => $_ }, @files ],
);
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could have been written

my @files_param;
for (@files) {
   push @files_param, { NAME => $_ };
}
$template->param(\@files_param);
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