During fusion 4 hydrogen nuclei (protons) become one helium nulceus (alpha particle). this causes the sun's core to slowly become more compact and therefore hotter, increasing the rate of fusion. 4 billion years ago the sun was 30% dimmer than now, but the Earth was still quite warm (although I disagree with the opinion that the temperatures were close to the boiling point) because there was a lot of CO2 (though a lot less then what it started with. The Earth started out with as much CO2 as Venus, but most of that CO2 had become limestone by 4 billion years ago, leaving a Nitrogen-CO2 atmosphere slightly thicker than the modern atmosphere). As the sun slowly brightened CO2 was slowly taken out of the atmosphere by forming limestone. When it warms up the rate of chemical weathering of rock increases, leading to more calcium ions in the oceans, increasing the rate of limestone depostion relative to the amount of CO2 released by volcanos. similarily, a cooling trend would reduce the weathering rate, allowing more volcanic CO2 to stay in the atmosphere. This is the Earth's most important negative feedback loop, it is what keeps Earth's average temperature within a narrow range.
The pricipitous drop in CO2 levels in the Devonian and Early Carboniferous is associating the the rapid expansion of terrestrial plant life and therefore the start of the soil as a carbon sink. Also, respiration from plant roots pumps CO2 into the groundwater, increasing it's acidity and, therfore, ability to chemically weather rock, making limestone depostion more efficient.
Oh, and one shouldn't take that chart's CO2 levels as dogma, I've seen a similar chart on Wiki (one of the images here:
http://en.wikipedia.org/wiki/Paleoclimatology ) showing lower CO2 levels during the Mesozoic (1000ppm instead of 2000ppm). the orange line is the same data as in the chart in the OP, the data I most agree with is shown on the green line. The dark blue line is the consensus estimite. The data shown by the purple line looks like nonsense. I also don't like that graph because it simplifies the climate data way too much.