Can you please explain the comment about nature compensating for part of the emissions so "only about half of the CO2 we emit stays in the air"?  Through what mechanism(s) is this happening?  Does this mean that humans are emitting a net 3 gigatonnes of CO2 which is not handled, or is it already included in the 6 gigatonne figure?  Thanks!

That's just straight math from observed numbers. We know pretty much how much carbon we have burned since the beginning of the industrial era (270 Gton of carbon); which leads to knowing how much carbon dioxide we have added to the atmosphere (270 Gton of carbon). We know how much the carbon dioxide in the atmosphere has actually increased: (160 Gton of carbon). It therefore must be that natural sinks have absorbed 110 Gton of carbon in the past 200 years, over and above the pre-industrial source/sink balance. Exactly which natural sinks have sucked up the excess carbon is still a matter of debate, but the amount is determined by the math.

In fact, the IPCC model, to be conservative with respect to human effects, assumes that there has also been an increase in natural sources, at present unknown, of carbon dioxide of equal magnitude to human sources over the same period (i.e. another 270 Gtons of carbon over the preindustrial amount), which in turn mathematically requires that presently unknown natural sinks have absorbed an additional 380 Gton of carbon, or 120% of what humans have produced, more than their preindustrial amount.

If in fact these presently unknown sources do not exist or are smaller, then the effect of human activity must be larger than the IPCC estimates. Of course, if they are larger than assumed, then the effect of human activity will be smaller; but it's increasingly unlikely that such large unknown natural sources of carbon dioxide exist, particularly ones that have increased their activity in parallel with human output, by coincidence.

The comment about nature compensating for part of our emissions could refer to two different things: the recycling effect of the biosphere (photosynthesizers) or the missing carbon  phenomenon.  The biosphere is a major carbon sink and a carbon reservoir-- the ocean and forests both take up carbon for photosynthesis (plankton in the ocean are very busy). However, the biosphere is calibrated, if you will, for a carbon cycle that doesn't involve massive emissions from stored carbon resources (fossil fuels),  so it isn't really equipped to handle extra carbon.
Since we have a good idea of how much carbon we've emitted since the Industrial Revolution (money makes people keep records), we have a pretty good idea of how much carbon ought to be in the atmosphere right now--and it isn't there. There is no certain answer at this point about where the carbon actually is; there are only three places that it can be hiding logically.  The ocean; the terrestrial biosphere; or the atmosphere (and we know it's not there).
The greenhouse effect is causing a 'greening' at higher latitudes--plants that couldn't survive at high latitudes because of colder temperatures are now able to move there because the temperatures are right for them--but that is unlikely to draw down enough CO2 on its own to offset that 7 Gt per year of anthropogenic CO2 emissions.  The ocean has a much faster turnover rate for CO2, so a number of people think that it might be there, as a dissolved gas.
In short, we know we're emitting about 7 Gt of CO2 per year (about 6 Gt in fossil fuel burning, and 1 Gt from deforestation),  but only 3.5 Gt appears to stay in the atmosphere.