Bought more storage, iPhone not using it?

I bought 20GB of iCloud storage last night and backed it up today but still it won't let me even take a picture or get a new app because "there's no storage." That doesn't make sense. What do I do?

Added (1). Do I have to manually move things into the cloud or will it do it automatically?

ICloud is a remote cloud-based storage solution. You may have to make backups of your current stuff, and then remove the apps and content you don't really need locally on your phone. By default it should do regular backups automatically unless if you have set it to backup manually to your computer with iTunes.

Note: remote cloud-based or external storage does NOT replace internal storage in mobile devices as it's not always available (requires a constantly reliable Internet connection) and too slow for the average I/O operation (loading apps and other system files).