But now how would I get it to only print out the entry once even when there are multiple matches
Like so:

#!/usr/bin/perl 

   use warnings;
   print "Search for what: ";
   chomp(my $search = <STDIN>);
   my %found;  #used as a set later
   @words = split(/ /, $search);
   open(DATA, "books.txt") or die "error opening file $!";

   while (<DATA>) {
           foreach $word(@words) {
                   chomp($word);           
                   if (/$word/i) {
            print unless defined($found{lc($word)}); 
            $found{lc($word)}=1; #indicates that word has been found
                                 
                   }
           }
   }
[download]

Update: This prints the line for each word only once. If you instead want to print each line only once, regardless of how many words it matches, try this:

#!/usr/bin/perl 

      use warnings;
      print "Search for what: ";
      chomp(my $search = <STDIN>);
      @words = split(/ /, $search);
      open(DATA, "books.txt") or die "error opening file $!";

      while (<DATA>) {
              foreach $word(@words) {
                      chomp($word);           
                      if (/$word/i) {
                           print;
                           last; #skips checking this line against oth
+er words         
                      }
              }
      }
[download]