To understand frames per second, you have to understand frames.
When you're watching a video or playing a game, you're not watching actual movement -- you're seeing a number of still images. 24~30 still images per second is roughly the number required for the average brain to fool itself into thinking that it's seeing actual movement on a typical sized TV with you sitting a typical distance away. It is NOT the maximum that your eye or brain can see and process!
In order to understand the difference between 60fps and 30fps, let's forget about frames per second for now and just go with frames:
Let's say a friend uses an uber camera and takes still photos of you moving from a sitting position to a standing position.
If he shoots 2 frames, you'll have 2 still shots between your sitting and standing positions.
If he shoots 30 frames, you'll have 30 still shots between your sitting and standing positions.
If he shoots 60 frames, you'll have 60 still shots between your sitting and standing positions.
Now, if the total time it took you to go from that sitting position to a standing one was exactly 1 second, and you process your still shots into video to match real-time, you will have 2fps (super choppy), 30fps (pretty smooth), and 60fps (very smooth), respectively.
The faster/slower issue comes up when, say, you only took 30 shots but process it into 60 frames per second. At that point, your entire movement from sitting to standing will be over in 0.5 seconds instead of 1 second. It's not an issue with most newer games because they'll work with hardware to create or skip frames to match the developer's desired "realtime" timings.
The technology of showing you this fake movement isn't different between TVs, computer monitors, and movie projectors. They all show you a number of still frames each second, not actual movement.
And this is why high-performance gaming hardware is in demand. When given a lot of graphical details to process, slower graphics cards and CPUs won't be able to process as many still frames per second as required by your eyes and brain to detect the images as fluid movement.
15fps is fine on old phones with small screens, but that same 15fps will be choppy on even a 17" monitor, as there are big gaps between still images that your brain notices. It'll think it's being bombarded by myriad still images vs. seeing smooth movement, thereby inducing fatigue and nausea.
Obviously, some people find 30 frames per second to be fine. But they're really wrong when they say that people can't tell the difference between 30 and 60. Maybe between 50-60fps, or between 60-100fps, but 30-60 is a big difference that the average human brain can differentiate quite easily.