Why specialization is important in the operation of a market?

This is one of the clearest lessons of classical microeconomics. I'm not sure I can show it graphically with the tools on this site, but Sal Khan has a fine explanation on Khan Academy, here.

Let me try to explain here without graphs, using a simple example. You and I will operate a firm to produce cookies. The firm has two tasks,

1) bake the cookies (or just, "bake")
2) package the cookies (or just, "package")

You would not expect to have the exact same productivity as me, so let's also assume the following:

You: Bake 40 cookies per hour
You: Package 120 cookies per hour

I: Bake 10 cookies per hour
I: Package 30 cookies per hour

Let's assume we have an 8 hour day, and that we must package all cookies that we bake during the same 8 hour shift.

If you work by yourself, you would produce 240 cookies -- 2 hours baking and 6 hours packaging. If I work by myself, I would produce only 60 cookies -- 6 hours baking and 2 hours packaging. If we work separately, we produce 300 cookies.

However, if we specialize, we can produce more together. From the example, you can see that you are relatively better (faster, in this case) at baking, and I am relatively better at packaging. You can bake 420 cookies in 6.5 hours. I can package 240 of those in 8 hours, and you can package the remaining 180 cookies (the ones I don't have time to package) in the 1.5 hours that you do not spend baking the cookies.

In a more practical example, a firm would probably hire and staff to achieve the right ratio of bakers like you (probably at higher wage rates) to packagers like me (probably at lower wage rates), so that the number of cookies baked by the baking department would equal the number of cookies packaged by the packaging department in any given period of time.