A 1-year zero coupon bond has a yield of 6%. A 7-yr zero coupon bond has a yield of 6.25%. Confirm that 6 year forward rate starting 1...

The key thing to understand here is what we're talking about when we say "forward rate".

The yield of a given bond is fairly straightforward; since these are zero-coupon bonds, they don't pay out anything until they mature, at which point they will pay out the original value plus their yield. For the 1-year bond we can just read this off; 6% yield means that we'll receive 1.06 times the original value when the bond matures. Yields are calculated on a per-year basis (APR, "annual percentage yield"), so the 7-year bond will pay out 1.0625^7 = 1.5286 times its purchase price.

The forward rate is a somewhat trickier concept, however; it's the expected yield that a bond should have if we buy it at some point in the future. In order to price the forward rate, we use the assumption of zero-arbitrage; that is, we assume that there's no way for someone to make guaranteed money simply by buying some bonds and selling others.

This means that we have two portfolios to consider.

Portfolio A is just buying the 7-year bond, paying 6.25% APR.
Portfolio B is buying the 1-year bond, then a year from now when it matures, buying a 6-year bond at whatever the going rate is.

By the zero-arbitrage assumption, these two portfolios should pay out the same amount in 7 years. The rate we expect the 6-year bond bought a year from now to be is the forward rate we're looking for; it should work out so that we make the same amount of money in portfolio B as we did with portfolio A.

For simplicity, let's assume we invest $1000. (The amount you choose is arbitrary, simply for convenience. You could say X or something, but I think it's easier to visualize with $1000.)

The 7-year bond will yield 6.25% each year, compounded annually, so after 7 years we'll have:
$1000 * (1.0625)^7 = $1,528.63
We're done with portfolio A; it gives us $1,528.63 after 7 years.

The 1-year bond will yield 6% the first year, so after 1 year we'll have:
$1000 * (1.06) = $1,060.00

To make portfolio B match portfolio A, we need to buy a 6-year bond with an appropriate yield so that this $1,060 ends up becoming $1,528.63. Call the rate we need r:
$1060 * (1 + r)^6 = $1528.63

Solve for r:

(1 + r)^6 = 1.4421
1 + r = 1.0629
r = 0.0629

Thus, the interest rate on the 6-year bond needs to be 6.29%, so that portfolio B will pay out the same as portfolio A. Thus, the forward rate must be 6.29%.