How is the internet able to transmit data so fast?
Perhaps this would be better as a comment, but it is too long. I wanted to address the fact that your statements indicate that you perceive the data to be transmitted almost instantaneously, but none of the observations you present actually prove that it is transmitted quickly at all.

You mention videogames. Game developers are WELL aware of the fact that there is significant latency between players, so they pull off several tricks. One of them is that they have the client guess about certain information that it hasn't yet received from the server/other players. For example, your client knows your opponent's position and velocity from, say, 50ms ago. So it extrapolates and says, "if his motion is like this, he's probably about here now", and you see this predicted position. Most of the time, it's pretty accurate (probably because of the effort programmers put into this), and it actually feels like there's no latency. Other times it's inaccurate, and to you, it appears that the player was shot, when really he was at a different position than what your console predicted.

On a related note, most displays run around 60Hz, and some do something called double-buffering. I won't go into the details here, but this introduces up to 33ms of latency between when the processor renders a frame and when it's actually displayed. Most people don't notice this, so I think it's reasonable to suggest that even if the network latency were equivalent to 33ms, you might perceive it to be instant, even without any programming tricks. 

In summary, the internet is not necessarily "fast". But smart people do smart things to make it appear as if it's faster than it really is. If you want more information, you may consider asking the people on gamedev.stackexchange.com.