I get asked this pretty often. Usually it’s because someone got a new phone or had to wipe their phone but sometimes, even though the Windows Phone backup/restore is amazingly full featured, the reinstalled apps don’t see to be aware of their licenses. Assuming you logged into your new Windows Phone with the same information, you are still entitled to all the same apps and in-app purchases you made with your previous phone.
There are two main methods I suggest.
Option 1 - Re-download
Basically, you click the Buy again. Having already purchased it and assuming you've logged in with the same information you did with your prior phone, instead of prompting for money, it should say something like, "You already purchased this." I found this one scary myself because it really sounds like it's going to charge you again. Also, if you’re reacquiring many apps, this might be a little tedious.
Option 2 - Use the website
The other way requires the use of a desktop/laptop computer but is more clear about the fact that it's re-installing an already purchased app.
On a desktop computer (that is, not using your phone's browser):
- Go to: https://www.windowsphone.com/en-us/my/purchase-history?signin=true
- Sign in if you haven't already (there should be a link to do so in the upper,right).
- Here you will see the apps you've purchased. You can click "reinstall" to have your phone reinstall them.
As I always say, please let me know how it goes.