If the screen were a mirror, you would see a reflection of the point source and there would be no intrinsic blur; but since you're looking at a screen which scatters light, you see the distribution of intensity at the screen. The image you see on a screen - whether the image is blurred or sharply focused - is the distribution of light intensity on the screen.
You can think of an object - or a real image of the object, as a collection of point light sources. When light from a point light source is brought to a focus on a screen, light intensity from that point source is all concentrated at one point on the screen.
But when light from a point source is not brought to a focus on the screen, it is spread out over some region on the screen. When you look at the screen, you see light (from that point source) over that region instead of at a single point; and you see it as a blurred point.
A real image can only be seen if its rays enter your eye, which means that you need to be on the right-hand side of your diagram, looking back toward the lenses, to see the real image. You can't see the real image if there is a screen intercepting the rays. If there is a screen, you can only see the light intensity at the screen -- which is blurred unless the image is focused precisely on the screen.
If the screen were, for example, lightly scratched plastic, and you looked through the screen back at the real image, you would see both the real image and a blurred image because some of the rays would go straight through the plastic without being scattered (allowing the real image to be seen), and others would be scattered at the plastic.
Best Answer
If you get in line with the light travelling through the lens at a location beyond the real image point and look back toward the lens, you can see the image "floating" in the air. If you get far off-axis, you won't see it. The light from the image must travel (somehow) to your eyes. It takes some practice to get your eyes to focus on the image rather that the lens or other surroundings, but it can be done. Mirrors work very nicely, too, to let you see images from off-axis positions.