In the case of the lens, you see that the the light rays first go through the lens, then the focal plane, and then through the plane of the real image. So if you're standing in the focal plane, the photons hit you before they converge in the real image: you're supposed to stand on the far right side! ;-)
So if you're standing at the focal plane and look to the right, you see nothing because the light only comes from the left. If you're looking to the left, you are seeing convergent light rays. An eye isn't really ready to focus convergent light rays. It's ready to focus parallel rays (from distant objects) or divergent rays (from nearby objects). For an eye, convergent light rays are like rays from an object that is "further than at infinity" and only a far-sighted eye will be able to deal with it. Otherwise whatever the eye lens will do, a human will see a fuzzy picture.
But when I say "fuzzy", it just means that the location on the retina depends on which direction of a photon from O you trace. If you only trace one particular photon, you will of course see one particular pixel through the retina. However, because the picture is fuzzy i.e. the activated retina pixel depends on the direction of the photons you trace, you can't talk about "apparent size" because to talk about the size, you need particular points of O to be associated with particular places of the retina which ain't the case if the picture is fuzzy (and if you pick particular photons from points of O, there's still no way to define which directions you mean for all points of O simultaneously).
The situation is analogous if you're in the focal plane of the convex mirror. You still see convergent light rays.
The eye will see a sharp final image as long as the final image is between infinity (real image in focal plane of eyepiece) and the near point (25 cm) of the eye (real image between $F_{\rm e}$ and eyepiece).
Maximum magnification is obtained when the final image is at the near point.
However it is often the case that the final image is made to be at infinity to reduce eye strain.
Best Answer
Your eye is a second optical system.
It re-focuses the diverging rays to produce a real image on the retina.
This process is exactly the same thing it does when looking at a nearby (i.e. not at effective infinity) object.