Give a persuasive explanation of why the carbon cycle is so important to our planet.

No problem! The carbon cycle is important because without it there would be no plant life on earth. Plants breath in CO2 and create oxygen, while animals work in the opposite, using oxygen to breath and exhaling CO2.

Carbon dioxide is present in the atmosphere in relatively small quantities but its cycling between the atmosphere and the earth is vital to life. It's taken up by plants during photosynthesis, converted into carbohydrates, transformed into animals by their ingestion of plant matter and released back to the atmosphere through plant and animal respiration. Fossil fuels are preserved plant and micro-organism remains and therefore contain carbon, when they are burnt this is released as carbon dioxide. The oceans absorb 25 percent of the carbon dioxide released into the atmosphere.
When the oceans absorb the carbon dioxide, some of it is used by the sea creatures to make calcium carbonate shells. These in turn,along with carbonate-rich sediments, can be turned into sedimentary rock. Carbon dioxide is released into the atmosphere by the weathering of these carbon-rich rocks such as chalk and limestone,
the carbon dioxide is one of the greenhouse gases and it is responsible for keeping the air near the surface warmer. Otherwise the average surface temperature of the earth would have gone down by 33 degrees celsius. The average temperature would have been -18 degree celsius instead of the present value of 15 degree celsius if the greenhouse effect is not there. Moreover carbon dioxide is taken by the plants and instead oxygen is released by them which is essential for human respiration.