Lets see if I can explain this graphically.
Have a look at the digram below,
These two "pictures" where taken at the same focal length at the same distance from a subject. The black vertical lines indicate the aperture opening. The red lines are sight lines from the top and bottom of the aperture opening. As you can see, they approach the subject at a shallower angle. When they meet (or get close to meeting) - they show a focused subject.
In the bottom graphic they are closer longer - thus, a longer depth of field.
Now, I know what your thinking - they only meet at one point in both situations. This is true. There will only be one point in the photograph that will be exactly in focus. However, your brain has a tolerance.
If you notice the lines are closer together in the bottom figure than they are in the top for a longer time. Your brain interprets "close to being in focus" as "being in focus"... again, with a tolerance. Your eyes just don't have the precision to detect see the minor loss of focus - so it still looks in focus.