Why are "transfer payments" not part of GDP?
Transfer payments aren't included in GDP to prevent double-counting. 

The reason the author's question is troubling you is because the answer externalizes everything that happens after the payment has been made (which is when the money from that payment DOES get factored into GDP, see below). I assume it does this to reinforce that transfer payments aren't counted in GDP, but you are right to question what's going on here. 

That \$100 million is transferred from the government to consumers, who will presumably spend a good portion (say, \$98 million) of those funds on goods and services, at which point those funds would be added to GDP. If we had included the transfer funds in GDP, the change reflected would be \$198 million instead of the more accurate \$98 million.

You can also have an example on a much smaller scale: 


You give your child \$5 to buy a burger; none of that is added to GDP. 
Your child spends \$4.75 on the burger; \$4.75 is added to GDP.