Yes, the world is literally getting greener, largely thanks to anthropogenically-elevated atmospheric CO2 levels.
Is the world getting greener?
Here's a map:
Source: Greening Earth: Spatial patterns, by Prof. Ranga B. Myneni, Boston U. (2015)
This excerpt is from a 2009 National Geographic article about how even the "Sahara" desert (really the Sahel) is greening:
Images taken between 1982 and 2002 revealed extensive regreening
throughout the Sahel, according to a new study in the journal
Biogeosciences. / The study suggests huge increases in vegetation in
areas including central Chad and western Sudan. ... "Before, there was
not a single scorpion, not a single blade of grass," he said. "Now you
have people grazing their camels in areas which may not have been used
for hundreds or even thousands of years. You see birds, ostriches,
gazelles coming back, even sorts of amphibians coming back... The
trend has continued for more than 20 years. It is indisputable."
Why is it getting greener?
An effect called the CO2 fertilization effect leads to healthier, faster-growing plants. Higher CO2 levels in the atmosphere cause increased plant growth. A study published in 2013 found that in warm and arid environments over a 28 year period (1982-2010), as CO2 levels increased by 14%, foliage cover increased by 11%.
For example, here are the results of an experiment of elevated CO2 levels on pine trees:
The fact that elevated CO2 is very beneficial for plants has been known to science for a century. The first F.A.C.E. (free air carbon enrichment) study that I'm aware of was done nearly a century ago, in Germany. Scientific American wrote about it in 1920. In that article Scientific American called anthropogenic CO2 “the precious air fertilizer,” and from this photo, which accompanied the article, it is easy to see why. It shows potatoes grown with and without the benefit of extra CO2:
Theoretical physicist, Prof. Freeman Dyson says:
about 15 percent of agricultural yields are due to CO2 we put in the atmosphere.
If you look at plant growth rates in response to CO2 enrichment & depletion, that conclusion is inescapable.
Yet 15% is really a conservative estimate, a lower bound. The agricultural yield enhancement from anthropogenic CO2 is really at least 15%.
Bunce (2012) finds that the widely-used FACE methodology for measuring plant response to elevated CO2 results in an underestimate of the agricultural productivity improvement from higher CO2 levels (as also explained by Prof. George Hendrey).
This video compares legume seedlings grown at two different CO2 concentration levels
Here's an article about a 12-year-long University of Michigan F.A.C.E. study which found that:
[...] CO2-soaked [birch, aspen & maple] trees grew 26 percent more than those exposed to normal levels of carbon dioxide.
CO2 is a more stable, less reactive molecule than O2. Venus and Mars both have atmospheres which consist mostly of CO2, with almost no free oxygen. But on Earth the opposite is true. There are more than 500 times as many O2 molecules in the Earth's atmosphere as CO2 molecules. Why is there so little carbon dioxide, and so much free oxygen, in the Earth's atmosphere? The answer is that Venus & Mars are dead planets, and the Earth is not. Its plants use CO2 and produce O2, and its animals use O2 and produce CO2. In the tug-of-war between plants and animals, the plants won. Plants have tugged the CO2-O2 tug-of-war rope all the way to the end. (Some carbon also ends up sequestered by animals in calcium carbonate shells, etc.) Animals are relatively scarce, compared to photosynthetic plants, and the plants have used up nearly all the CO2. The outnumbered animals just can't produce enough CO2 to keep up.
That's why, on Earth, O2 is plentiful, and CO2 is measured in parts-per-million.
Higher CO2 levels are very beneficial for agriculture. C3 photosynthesis plants, like wheat, rice, all vegetables, all nuts, and nearly all fruits, benefit the most from additional CO2. But even C4 plants, like corn and sorghum, grow better with higher CO2 levels, especially when under drought stress. Thousands of studies have found that nearly all plants benefit from additional CO2.
That's why commercial greenhouse operators use CO2 generators to artificially boost the CO2 levels in their greenhouses to 2.5x to 4x normal levels: because higher CO2 levels result in much faster-growing, healthier plants.
The IPCC's AR5 estimate is that 45% of anthropogenic CO2 emissions remain in the atmosphere each year. They estimate that 55% of total emissions are taken up by oceans and biosphere, and they think over half of that is probably taken up by the biosphere, through greening. AR5 estimates that the terrestrial biosphere removes about (2.5/9.2) = 27% [p. 6-3] or 29% [Fig. 6.1] of anthropogenic CO2 emissions from the atmosphere, each year, and that the oceans remove another 26% [Fig 6.1]. (There are wide error bars on those numbers, but the 55% sum has narrower error bars than the two addends.)
BTW, I also recommend to you Matt Ridley's 2011 Angus Millar lecture at the RSA, entitled, Scientific Heresy.