Now think about it from the investors side. Imagine there are a group of wealthy individuals.
Now imagine that interest rates were low and heading down, so that you got ~0% to ~3% returns in bonds, securities, treasuries, TIPS, money market accounts, CDs, savings accounts, etc. How many of those investors will choose one of those safe options versus how many would invest in new/expanding businesses including new product lines, better factories, etc? Probably most will choose to expand businesses since the interest rates are so low. Heck, they can even borrow money at that low rate to expand business even more. Suddenly there is a lot more production, more supply, and given the same demand, prices must fall.
Now imagine that interest rates are near 10% or higher. Bonds and all those other similar investments pay similarly high amounts. Will some of the investors take that route instead of investing in business expansion? Yes. Will investors borrow money for business expansion? Some may, but a lot fewer than in the case above. That means less expansion, less supply, and given the same demand, prices must rise.
From the investor side, prices and interest rates go hand in hand.