Why do bonds fall in value when interest rates rise? Well, to be precise, the prices of existing bonds will fall when interest rates rise. The prices of newly issued bonds are fixed. But let's back up a bit.
Higher interest rates mean that companies are likely to borrow less, produce less, and thus earn less. (This is because higher interest rates make it more expensive to borrow.) Since stock prices are tied to how much a company can earn, higher interest rates theoretically cause stock prices to fall. This, coupled with rising bond interest rates, makes bonds more attractive to investors.
Imagine 5% bonds with 10 years left until maturity that originally sold for $1,000 each. If you buy these bonds now, you'll be getting $50 per year from each of them and then $1,000 at maturity. But if interest rates have risen since those bonds were issued, and you can buy new 10-year bonds that pay you 10%, that amounts to $100 per year per $1,000 invested. You would obviously be willing to pay more for the 10% bonds than the 5% ones. So, the price of 5% bonds will fall. It will fall to the point where $1,000 invested in the 5% bond will bring you same total yield-to-maturity as $1,000 invested in the new 10% bond.