How does myopia actually work?
The image you gave is not clear because it does not show the fact that the divergence pattern of light from a close object is different from the divergence pattern of light from a distant object. 



Light rays from a close object (first image) diverge, and therefore they are in focus for the myopic eye.

Light rays from a distant object (second image) is approximately parallel, and therefore they are not in focus for the myopic eye. The lens corrects for this inability to focus and allows the distant object to become in focus.

Also note that the myopic person is still capable of focusing on close objects despite the lens because the lenses are still capable of dynamic adjustment, but less so than a person with perfect eyesight. You can test this by focusing on a near object, then putting on your glasses. The object should be initially out of focus, and then become in focus as the eye lenses adjust to the distance. 

Myopia/presbyopia is fundamentally an issue of the focal range of the eye becoming limited to a specific range. If the ability of the eye lenses to dynamically adjust is further weakened, it may require progressive lenses or bifocals to correct the issue.