Before your workout

 Stretching brings oxygen into your muscles. Without oxygen your muscles are cold, that is why stretching is referred to as "warming up" you muscles. The warmer your muscles are before your workout the happier they will be.
 If you are working out on cold muscles, they can't absorb shock. They don't respond like they should on impact. There are a lot of things that can be affected by this.
 You will get fewer injuries if you stretch before your workout. When your muscles have oxygen they can respond and they know what to do with the exercises you are performing. Without warming your muscles can't do what they need to.
 You will burn more fat if you stretch. If you ask me that is reason enough for stretching. With your muscles warmed up they are able to break down the fat cells instead of letting them collect where they land.
 You will have improved coordination with stretching. Your muscles are loose and so they can more better. Plus with the extra blood flow the blood increases in our whole body, this means that your brain is getting more oxygen and you can think better.
 Stretching protects your back from being injured. You back is something that you need to take the most care of. Stretching makes your back more able to take the stress that a workout gives to it.

After your workout

 When you have finished pushing yourself it is time to let your heart slow down. If you just stop exercising then your heart is going to fast for the rest of your body. This can make you feel weird after your workout and puts extra stress on your heart. Stretching gives your heart that slowing down time that it needs.
 Stretching relaxes muscles after a hard workout. Your muscles tense up while they are being pushed and worked. Stretching gives you a chance to work out all of that tension and end your workout with a loose body.
 The soreness of your workout will be much less intense if you take some cool down time to stretch. With your muscles being loved after being worked so hard your body will thank you.

The why behind what everybody tells you makes it easier to do. Now that you know the benefits that come from stretching, maybe you will find the time to work it into the beginning and end of your workouts.

Be sure that you are stretching all of your muscles, not just touching your toes. There are muscle groups all over your body, find them and stretch them out too. Stretch your neck, your back, your shoulders, your arms, your wrists, and all of your fingers. Stretch your torso, your hips, your legs (there are lots of muscles there to remember), your ankles, and each of your toes. Even your fingers and toes get worked and need a good stretch.

Remember not to bounce when you are stretching. Your stretches should be fluid motions, smooth and even. They should be slow and they should push you, but they shouldn't cause grueling pain. Make sure that you aren't doing damage when you are stretching out.