How do I use a Chinese spoon?
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.

EDIT:

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.