I'm not an expert on plants (I'm an engineer) but what I can tell you is that CO₂ is the most common and abundant carbon based gas in our atmosphere.

If we lived on a different planet, where that were not the case, then plants may have evolved to use some different type of gas. But as it stands, here on Earth, conditions were right for plants to use CO₂ in their growth. Now how exactly CO₂ is used by plants is a question that I myself would like to ask a biologist, but the fact that plants do use CO₂ is because of the environment in which they evolved.

CO₂ is by far the most common gas in the earth's atmosphere that contains carbon - the next most common (methane) is almost nonexistent by comparison. Plants use what they can get a hold of, and that is CO₂ because it's so common.