That fact that lenses focus images of objects follows from the basic diagram you already show:

Think of this as the definition of what a lens does. It also happens to show it focusing the image of something at infinity to a single point.

Now imagine you have a object at some finite distance and position relative to the lens. You can use the few special cases of rays you already know about to find where the lens will focus the image of that object:

The square at left is the object. We already know from the previous diagram what will happen to two of the rays from that object in this cross-sectional view. The ray going parallel to the center line will be bent to pass thru the focal point at right. The ray thru the left focal point will be bent to go parallel to the center line. Where those two rays meet at right is where the image of the object will be focused.

The math for calculating where the other rays from the object will go after being bent by the lens gets more complicated, but they converge at the same point for a ideal infinitely thin lens.

Of course real lenses aren't infinitely thin, so there are various approximations and tradeoffs in real lens design. The further the focused image gets from the center line, the less this ideal approximation holds. This is why real lenses have a maximum sensor size they are specified to work with. It is also why characteristics generally get worse at the edges of pictures.

Note that the rays getting bent in the center of the lens is a convenient simplification. They actually get bent at the two interfaces. When taken from far enough away and the lens isn't too fat, the bent at the center simplification applies well enough.

Rays are actually bent at the air/glass interface due to the different index of refraction of the two materials. Since the index of refraction changes as a function of wavelength, the focal points of a lens are effectively in different places for different colors. Of course we choose materials for lenses that have relatively flat index of refraction across the visible light spectrum, but there is no perfect material. One of the reasons for multi-element lenses is to use different materials with different refraction indexes to cancel out the variations.

The simplified view of a lens is easy to understand and requires only high school geometry. Real lenses get very complicated.