Probably tr isn't what you want here, and there's no need to capture the word, so no parentheses. A simple solution is:

my @count = $str =~ /\bthe\b/gi;
print scalar @count, "\n";
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The idea is that g searches globally while i makes the search case-insensitive. Assigning the results to an array means that the @count is populated with all of the successful matches. Placing the array in scalar context will return the number of matches.