How to increase storage space on my 64GB iPhone?

I recently received a message saying my iPhone had Low Disc Space. I went into storage and signed up to increase my storage space by paying a small extra sum which appeared to have started and was taken from my credit card. A short time has passed the storage showing did not increase and stays at 64GB and I have received the same message Low Disc Space and cannot store any extra photos.

What is the problem

[Re-Titled by Moderator]