Here is an excellent video explaining how the medial arch of the foot works and then illustrating how that affects proper gait.
The arch of the foot is supported by muscles and ligaments and over time can start to fall as we age. Some people have had flat feet their whole lives. Most people will look to orthotics to support their flat feet or low arches, but a passive support like an orthotic arch does not go after the key problem: weak foot muscles and poor gait. Strengthening these foot muscles can help to rebuild that arch without the aid of lazy footwear. This may take a year or two to accomplish, but the benefits will last a lifetime.
The following videos show a few exercises for strengthening the medial arch of your foot:
The following video shows you the towel scrunching exercise to strengthen the general intrinsic muscles of the feet. You might find it more challenging if instead of simply contracting and relaxing, you try to scrunch ALL of the towel into a ball
It is also important to strengthen some of your calf muscles. The tibialis posterior and tibialis anterior muscles do the bulk of the work lifting your arch. Strengthening them will help to support the arch. As an added bonus, these exercises will also help with ankle stability issues. This video shows how to strengthen the tibialis posterior muscle:
There are some good videos on strengthening the tibialis anterior muscle on this post.