I have been having this problem too on my new iPhone 6s Plus.
I haven't found a fix but I have found a workaround which is to tap the "installing" button to pause it (the one that has the circle going around it to show installation progress), give it a few seconds and then tap it again to unpause. I've had to do this in some cases 10 times but eventually the download will work.
Things I've done to try troubleshooting this:
(1) At first I thought it was my wifi or some networking issue, so I used Settings|General to reset the wifi but this had no effect
(2) I ran speedtest.net and it produces roughly the same result as my iPad which is about 70MB/s download and 3MB/s upload
(3) I've had no trouble using the browser, siri or any other networking related software on the phone
(4) I put my ipad and iphone 6s plus side by side on the table and observed the ipad download speedtest.net from the appstore very quickly and the iphone 6s try to get started and appeared hung. So I repeatedly did the pause/unpause trick described above and eventually it got going and finished.
My best guess is that is the app store has to do something unique for the new iPhone 6s (possibly specific to the plus model?) and is having trouble.