To answer this question, you need to know how forward prices are derived : non arbitrage argument. Thanks to it, we will show that we necessarily have $ F_T = S_0e^{rT} $ where $F_T$ is the price of the forward for delivery at time $T$, $r$ the risk-free continuous interest rate and $S_0$ the current price of the asset.
If $ F_T > S_0e^{rT} $:
I enter in a forward to sell the asset. I borrow the equivalent of $S_0$ at time 0 (and so I owe $S_0e^{rT}$ at time $T$), I buy the asset and wait for the delivery. At maturity I get a riskless profit of $ F_T - S_0e^{rT} $. As everyone will do this trade, the price of the forward should decrease and the price of the asset increase. Thus, the arbitrage opportunity will disappear.
If $ F_T < S_0e^{rT} $:
In this situation, I can borrow the stock, sell it (for $S_0$) dollars) and gain the risk free rate on the cash ( I will get $S_0e^{rT}$ at maturity). In the same time, at time $0$ I enter into a forward to buy the stock for $F_T$ dollars at time $T$. So, at $T$, I use the cash to pay the forward $F_T$ as agreed and give back the stock to the person from who I borrowed it. I get the riskless profit $ S_0e^{rT} - F_T $. With the same argument than above, this situation should immediately disappear if it occurs.
Conclusion : You have this formula $ F_T = S_0e^{rT} $ that tells you rising interest rates imply rising forward prices. As it comes from a non arbitrage argument, it is "meaningless" to find an economic reason to it except that : if this relationship does not hold, some persons could become infinitely rich without risks.