Not every rich person got an iPhone. The rich people without an iPhone did not had equal amount of less kids.
There are two groups, one has an iPhone, the other has not. The assumption is that two groups are big enough to have equal amount of people from any other group that can explain the decline in fertility, i.e. equal amount of rich/poor, educated, etc.
They can control for this because they know which people had access to the iphone based on the AT&T network coverage.
At the end of the abstract they state the likely explanation of this seemingly spurious correlation:
> National-survey evidence on time use and sexual behavior is consistent with the iPhone reducing in-person interactions, increasing pornography use, and reducing sexual frequency.
At the end of the abstract they state the likely explanation of this seemingly spurious correlation: > National-survey evidence on time use and sexual behavior is consistent with the iPhone reducing in-person interactions, increasing pornography use, and reducing sexual frequency.