The game checks your Internet connection, and if it's not stable enough to sync or the network response time is slow (your Internet can still be fast and good at the same time), it warns you about it.
Let's try the following steps:
1 - Restart your device. Press your device's power button for about 30 seconds, until your device restarts.
2 - Please make sure other apps that require Internet connection work fine.
3 - Make sure you have stable Internet signal (for example, you are not far from your modem if you are using Wi-Fi) and open the game. You may need to reboot your modem/router. Keep in mind that sometimes networks get overloaded or malfunction, meaning that the connection may not be stable regardless of what the indicator says.
4 - Make sure you are not running out of free space on your device. To check your free space go to Settings of your Kindle > Device Options > Storage. The game can't probably download required content to continue.