It appears they used the Forward Pricing Model first. Then, after solving for the Forward price, they used the Forward Rate Model but ignored the right-side of the equation as it pertains to the Spot rates. After that I can’t understand their algebra as it relates to the left side of the Forward Rate Model.
Use the Forward pricing model: subsitute F and solve for f(j,k). 
f and S are interest rates, not prices. F and P are prices (you can even think of them as PVs). 
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Crisp explanation.
It appears that k=1 at all times with forward rates since a forward curve discounts back one period. Am I correct?
Yes.
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Why is subscript (j+k) always in the numerator and subscript (j) in the denominator?
In the Forward rate model, if I multiply both sides by (1 + Sj) ^j, then the LHS will be the FV of the reinvestment strategy while the RHS will be the FV of the buy and hold strategy. A little bit of that trusty aljibber gets you to the numerator with (j+k) and the denominator with j.
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And a little bit of trusty proofreading changes the first instance of denominator to numerator.
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Fixed!!! 