Think about how you are able to see any object. Light from the object enters your eyes. Whether the object appears blurry or sharp depends on the ability of your eye's lens and cornea to bring the rays to a focus. It doesn't depend on the object itself.
Now, if some optical system takes light rays (as an abstract of the wave front normals) and redirects them so that at some point in space they take on the same pattern of paths (maybe larger or smaller, but the same pattern) as the original, the a screen placed at location will capture (and reflect) the pattern, i.e., the image. Your eye can capture the light rays from that pattern.
Next, think about how many different directions the light rays (an infinite number of them, approximately) from the original are spreading outward. Then think about the semi-infinite number of patterns of the original which could be created on some finite distant hemisphere by the mirror (in your situation). It's not merely on image directly in front of the mirror. It's a spherical/parabolic/ellipsoidal surface where the patterns matching the original pattern exist. And that location is where your eye will focus.
Do an experiment
You could do this trick with a camera. Get a camera which is manual focus and has distance markings on the focus ring. Stand in front of a plane mirror and measure the distance to the mirror. Focus on your mirror image with the camera. Record the focus ring distance. Do this several times. Compare the two values.
Now use a concave mirror with some object producing a real image on a screen. Position yourself so that the mirror and screen with image are in a line with yourself. Measure the distances to the screen and the mirror. Remove the screen and focus the camera on the image and record the focus ring distance. How does it compare with other distances?