Why would higher expected inflation increase corporate earnings and dividends' rate of increase?
I don’t have the book, but the passage hints that it will be covered in Chapter 13.

The passage has “reasonably project” as a disclaimer, which does imply that it would not hold in all cases. It is not that hard to find counter-examples. If we define inflation as a change in the CPI, oil price spike will normally raise CPI. The higher energy costs are bad for firms with energy input costs, while the oil producers that benefit may be foreigners.

However, if we stick to an idealised scenario, where all prices are rising at roughly the same rate, we can get the result that profits rise in line with inflation. If we make two assumptions:


real activity is not affected by the change in inflation rates;
profit margins are stable.


We then are in a situation where we are just projecting current variables forward with the same real growth rate, while nominal variables are growing by the sum of real growth and inflation. 

If profit margins are stable, then that implies that nominal profits are proportional to nominal revenues, and so would scale with inflation.

The passage implies that the author thinks those two assumptions are reasonable, and presumably detailed in Chapter 13. (I don’t have any references handy, but I’ve certainly seen similar arguments many times. The main difficulty is whether higher inflation is associated with higher or lower profit margin.)