Answer:
A simple way to explain it is:
When you have an infection and take antibiotics for it, the weaker bacteria are killed first, with the stronger ones surviving, or taking longer to destroy. So when you don't take the full round of antibiotics, the stronger bacteria are the ones left in your body, and they are the ones that will continue to multiply, resulting in a bacteria resistant to the antibiotic. This is why you should always take the full prescribed course of antibiotics.
Also, when antibiotics are prescribed, the body's natural defense system (the good bacteria) are destroyed along with the bad bacteria. This is why you should always eat yogurt with active cultures or drink buttermilk while on antibiotics - the active cultures put the "good" bacteria back in the body.