The Play Store asks me to pay for the app even though I bought it
Sometimes, the Play Store doesn't reload the purchase status of the app and asks you to pay for it again. Even though you'd never get charged twice for the same item and the Play Store would eventually tell you you already purchased the app and you'll get it for free when you press the purchase button, there are a few things you can do to force the Play Store to update the purchase status.
First, you can try rebooting your phone and re-launching the Play Store which usually helps.
Second, you can try going to Settings->Apps, select Google Play Store, select Storage&Cache and press both "Clear storage" and "Clear cache". This should force the Play Store to reload the up-to-date purchase status from the Internet.
In addition, please check this FAQ which can also be one reason why the Play Store asks you to re-purchase the app: