why do the prices of fixed coupon bonds and other fixed income securities rise when rates decline?

Answer 1

Because the rate on the bond in question does not change with the change in market interest rates (it's fixed for the duration of the maturity period subject to call provisions etc), the bond in question is therefore more valuable to an investor when rates decline because similar bonds are being issued in the market at a lower interest rate.

Answer 2

If I am getting 6% on a bond and the rates on other bonds fall to 5% then you will need to pay me more for the 6% bond so that I can go out and invest in a 5% and not lose on the transaction.