grep /$_/, [ @{$tbl{$i}} ]

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means:
Dereference the array-Ref in $tbl{$i}, make a reference to an anonymous array ( [ ] ) from the resulting list and grep through a list with a single item: the single reference to the anonymous array.

@tbl{$i} is btw. wrong, see this example : 
$perl
 perl 
use strict;
use warnings;

my %hash;
$hash{a} = [ 1,2 ];

print "@hash{a}", $/;
Scalar value @hash{a} better written as $hash{a} at - line 7.

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edit: several updates/corrections due to some misleading thoughts while typing too fast

In reply to Re^3: greping thru a hash of arrays by linuxer
in thread greping thru a hash of arrays by pglenski

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