In Asian restaurants the soup comes with a flat-bottomed spoon. It seems awkward to use this spoon as one would a "western" spoon.
Is there a special technique to using a Chinese Spoon?
What is the functional reason for having the flat bottom?
You would use the flat bottom spoon as you would use a regular spoon. The main different between the flat-bottom spoon and western spoons is as you can see the flat bottom and the fact that the flat bottom spoons are usually bigger and can hold more liquid. I'm not sure why it would be awkward to use it compared to a regular spoon. You don't need to stick the whole spoon in your mouth. There is a rounded edge around the spoon before it is flat. You put just the rounded portion of the spoon into your mouth and drink from it.
The reason for the larger spoon probably stems from the fact that many Asian soups have noodles and large chuncks of vegetables and meats in it. Using a western spoon, you would get some of the noodle or meat and have little room for the broth. But with the larger flat bottom spoon you can get the noodle, meat and still fit an appropriate amount of broth on the spoon.
Also note that it is common practice to use the flat bottom spoon and the chopsticks simulataneously while eating a bowl of noodle soup. One would use the chopsticks in their dominant hand (right hand lets assume). They would then use their left hand to use the spoon. As you have probably experienced in your life unless you are ambidexterous, using the left hand to do thing you normally do with the right can be hard. With the flat bottom, there is less chance of spilling than with a traditional rounded spoon.
The other answers haven't answered your first question about technique.
You hold the spoon with your index finger in the "groove" of the handle and your middle finger and thumb underneath:
This gives you a very secure grip and you scoop up soup by twisting your wrist.
The soup spoon the "western" equivalent of the Asian "flat-bottom" spoon. Both are used to lift liquid which is to be sipped. Think ladle.
They are different from "western" dessert spoons which are inserted into the mouth. Think fork.
The channel that runs from the large spooned section of the Chinese flat-bottomed spoon, is used to carry the soup/gravy to it's thinner end. Just lift the large end higher and watch the liquid run along the channel to the other end. This can be used to feed small babies and sick people more liquid food. You just drain the liquid this way and leave the solids behind. It is also an accurate way to transport the food without it escaping out both sides of the mouth. This channel is a very useful and often forgotten feature.
The Asian or Chinese flat bottom spoon not only has its application in holding more soup broth and/or liquid, its material, usually made of ceramic or porcelain tends to keep its handle cool even when immersed in a hot soup unlike the European/Western metal or silver soup spoons. The flat bottom also makes them easy to store as they are stackable. In addition, the ceramic/porcelain construction of the spoon is less likely to scrape the bottom or sides of a ceramic/porcelain soup bowls especially the elaborate decorative bowls. The Chinese cuisine has been using such spoons for (almost 3,000) years and its application and contribution is so important in many Asian tables. Use it as you would like a metal Western/European soup spoon.