What to see
On the right you see part of me while I'm making saccades, that is rapid eye movements between two fixation targets. These targets are my own eyes, which I am viewing in a mirror. The movement of the eyes is quite obvious, wouldn't you agree?
What to do
Now do this yourself – stand close to a mirror (nose nearly touching) and look to an fro between the eyes. Do you see you eyes move? I bet you don't! Although your eye movements will be similarly obvious as mine, you cannot see them. Have someone check – they will see it. Or take a “selfie movie”:
Many smartphones these days come with a front camera as well: So switch to front camera (the button to do this might look like on of those on the right) and observe the image (you don't need to record). Hold the phone rather close (<10 cm), and tilt the camera so both eyes are in view. Look to and fro between you eyes – and now you see your eyes move!
[An alternative to smartphone with front camera is a notebook with built-in camera, as used for video telephoning: Enlarge your own image (on Macs use “Photo Booth”), move close to the camera and look at your eyes.]
The interesting thing here is the absence of a perception, namely not perceiving your own saccades under normal circumstances. The mechanism behind this is called “saccadic suppression”. This mechanism is useful to suppress the motion blur during your saccades, which would be quite distracting, were it not for saccadic suppression. The suppression is not total (different studies differ in effect strength between a factor of 3 to 10), but it is good enough, obviously. The duration depends on the size of the saccade, 50 ms is in the right ballpark.
So why can you see your saccades in the camera selfie? Because the viewer shows the image with a little delay, so the eye movement appears while the saccadic suppression is already over.