People travel by air for mainly two reasons, and both require money:
Since the source of that chart is FAA, my answer is mostly related to USA, but it can generalized globally.
Years leading to 2000, there was an economic boom. Air travel was increasing. By 2000-2001 (before 9/11), dot-com collapse resulted in a recession. This forced companies to lay offs and not to pay for air travel.
Economy started to get better after that, and then the recession of 2007-2008 hit. Companies learnt an even bigger lesson not to spend money on luxuries.
Economy started to get better again. Air travel started to increase again, but did not peak and is not predicted to peak to the same level as of 1999-2000.
Why? After a financial calamity, companies and people are very reluctant to adopt same spending behaviors as before the calamity, since there is an uncertainty of it happening again.
After 2000, airlines was one of the industries which was hit the hardest. So they wanted to maximize their profits. That means that all airplane seats must be occupied. This resulted in fewer flights, and also increasing codeshares.
Many airlines acquired other airlines. In the US alone, several major airlines merged together, eliminating redundant and unneeded flights. This reduced air traffic too.
Also with economic issues, people started to spend less on their hobbies. As a lot of GA traffic occurs on airports with ATC, this reduced ATC operations too.
As Federico has mentioned, only A380 is the larger aircraft so this reason is not likely to contribute for decline in air traffic. On top of that, A380 is mostly used on international routes whereas the majority of air traffic is domestic.