Oftentimes, app download issues are caused by problems with Google Play's cache or data. Fortunately, there's an easy way to remedy them:
1. Go your device's Settings menu.
2. Go to Applications (may also be labeled Application Manager).
3. Tab over to All Applications.
4. Search for and open the Google Play Store app.
5. Tap on Clear Data and Clear Cache.
6. Relaunch Google Play and then download Swype.