The carbon has multiple roles that contribute to the diversity of life on the Earth, with the two most important being that it is the most abundant greenhouse gas in the atmosphere, and that it is used by the plants for their functioning.
The carbon is the most responsible gas int he atmosphere that enables the existence of the Greenhouse effect. This gas manages to trap the heat into the atmosphere, thus making the Earth much warmer than what it would be without it. That warmth enables the life to exist, develop, evolve, diversify across the planet.
The carbon is also crucial for the survival of the plants because they use it in the process of photosynthesis to produce their own food. Without the carbon that would not be possible, thus the plants would not exist, and if the plants don't exist than there will be no animal life as well.