When interest rates are higher, call option prices are higher and put option prices are low?
This is easy to see if you look at put-call parity: a higher risk-free rate makes the present value of the strike price smaller, so the call price has to be higher or the put price lower for the two sides to remain equal.
Higher interest rates mean higher call premiums.
Suppose you own 1,000 shares of stock X, currently trading for $100. Your capital commitment = $100 *1,000 = $100,000.
Instead of owning stocks, you can use call options to get the same exposure with lower capital. Suppose you can buy call options on stock X with a strike price of $100 for a premium of $40. Your capital commitment = $40 * 1,000 = $40,000. You still have exposure to 1,000 shares of stock X but now have $60,000 of cash that you can earn interest on. The higher the interest rate, the more valuable call options would become.
Higher interest rates mean lower put premiums.
Suppose you could short 1,000 shares of stock Y, currently trading for $100. Doing so would provide an immediate cash inflow of $100,000 that you can earn interest on.
Instead of shorting the stock, you can use put options on stock Y. Suppose you buy put options with a strike price of $100. When you eventually exercise the put option, you would receive cash inflow of $100*1,000 = $100,000 that you would earn interest on. But until that time, you are forgoing interest income. The higher the interest rate, the higher the opportunity cost of forgoing this income (another way to look at it: the higher the interest rate, the lower the present value of $100,000 that you would receive upon exercise of the put option). This would make put options less attractive and decrease their premiums.