When interest rates go up, there are two effects that explain the positive link with the increase in the price of a call option (according to Hull). There is the quote: " As interest rates in the economy increase, the expected return required by investors from the stock tends to increase. In addition, the present value of any future cash flow received by the holder of the option decreases".
The second explanation is the most intuitive for me. The present value of the payment is smaller (if you exercised the option). and so, basically you get the same thing (the stock) for a smaller present price.
By the way, the relationship between interest rates and stocks is not so straightforward. For a counterexample, look at the last years when the FED was hiking its interest rate and we saw a huge rally in the S&P 500. Interest rates increase due to increase in expected inflation and if this inflation is pushed by a limited supply, that means that companies are doing well regarding their capacities.