C02 does come back to earth with rain, carbonic acid, why rain is slightly acidic. Not to be confused with sulfuric acid rain from burning coal.
Also plants absorb CO2 as part of the photosynthesis cycle to make sugar/energy to sustain life. The equation is 6 CO2 + 6 H2O → C6H12O6 + 6 O2 (need to add energy, i.e. sunlight) Plants also have a cycle at night called the Krebs cycle that uses oxygen, but that would take a long time to explain and not germane to the discussion.
Carbon storage or carbon sequestration is what some folks are pushing for, would be to keep areas of slow growing perennial vegetation that would store carbon in the plant material for very a long time, multiple centuries in some cases, have a tree that grows for 300-1000 years, dies and the decomposition of the woody biomass which is rich in carbon takes centuries to decompose and as long as the soil isn't disturbed the carbon will stay locked up in organic matter.
If you plow soil rich in organic matter, much of if becomes oxidized and C02 is released into the atmosphere, and the overall level of organic matter decreases. If you harvest annuals for food, such as corn, beans, etc, the carbon is cycled back into the atmosphere as the consumer exhales CO2 in normal animal respiration. And the stubble left in the field will decompose into organic matter, but is released into the atmosphere as it oxidizes when the soil is disturbed.
Coal and other fossil fuels are rich in carbon because they are highly compressed organic matter, as they are released from the deep and consumed lots of C02 is released. C02 is a called a greenhouse gas because it acts like the greenhouse window in the atmosphere. Sunlight radiation enters the atmosphere, is reflected off the surface, then some of the radiation is reflected back to earth, so forth and so forth. If we didn't have any of the greenhouses gases, earth would be like Mars and without life. Too much greenhouse gases and the planet could warm like Venus and not be able to sustain life as we know it. There is a equilibrium to support life as we know it.
Now back to the article, yes they said the earth was getting greener, but I read it was getting greener in areas that had been void of vegetation due to extreme cold conditions or were previously covered in ice.
Now I'm not sure if man is contributing to climate change, or if it part of a cycle. We are releasing carbon that was stored from a extreme warm period many millions of years ago, but theories also figure the continents were more aligned in a mass group near the equator at the time also.