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Depends, there are different ways to think about it. There's the fact of the matter that you are 18% carbon by mass. How would you like having about a fifth of your body weight just thrown away?
But that doesn't really say why we evolved to rely on using it to construct ourselves. The real reason is because we literally are what we eat. It's the only way our body can create ourselves, that's the actual point of eating. All paths of things that are eaten eventually lead to plants, whether we eat plants directly or eat things that eat plants. So what do plants do?
Plants take energy from the sun, and take in moisture in the air (water) and carbon dioxide gas through tiny holes in their leaves. and uses this sunlight to combine the air into sugar. So you're made from carbon that plants turned from carbon dioxide gas in the air, it's essential that there was all this carbon dioxide floating around or we'd probably never be made out of it.
So what else? Well carbon is also just versatile. Carbon forms 4 bonds while Nitrogen, Oxygen and halogens form only 3, 2, and 1 bond respectively. So there are more ways to combine Carbon atoms than there are to combine other elements in the second period of the periodic table. I imagine this flexibility was also pretty important. I mean, just imagine a box of LEGOs, like would you rather play with only pieces that can only stack in one way, or would you rather play with pieces that can stack in multiple ways? Yeah, variety is more fun. ;P