Interest rates matter, especially to carry trades. Indeed, interest rate differentials are what determine which currency pairs are positive-carry or negative-carry. If you can purchase one currency that yields "X" in terms of another currency that yields "Y", so long as X exceeds Y (after costs) you make money. In the reverse situation, you lose money (since the differential would be negative).
Of course, interest rates are important for other reasons beyond carry trades. But interest rates, especially as they relate to carry trades, are important for U.S. stocks too. Modern financial markets, like the