First things first carbon is quite common element here in Earth. But it is not all, right? Otherwise we would be built from other, more common elements. The thing is in carbon reactiveness. Also energy needed to create carbon chemical compounds isnt that big when compared to etc nitrogen. It can bind up to 4 different elements (atoms). Thanks to this can givesingle, both and triple bindings! Carbon can give away electrons or take them making his degree of oxidation consequently from -IV to +IV. All this vareity leads to vareity of bigger elements that will be created from carbon later- carbohydrates, proteins etc. As life needs vareity to adapt to different situations and climax only carbon therefore can provide this.