Why is there a seasonal cycle to the amount of CO2 in the atmosphere?
Overall, the measured amount of $\mathrm{CO}_2$ increases, but in a cyclic pattern, as can be seen in observations shown below:



Image source: NOAA: Trends in Atmospheric Carbon Dioxide (numerous observation data sets and graphs are available from that website)

The fluctuations are seasonal, which is explained in the web page Seasonal carbon dioxide range expanding as more is added to Earth's atmosphere (NSF, 2013) summarises the seasonality as being


  Levels of carbon dioxide in the atmosphere rise and fall each year as
  plants, through photosynthesis and respiration, take up the gas in
  spring and summer, and release it in fall and winter. Now the range of
  that cycle is expanding as more carbon dioxide is emitted from burning
  fossil fuels and other human activities.


Specifically, the effect is more pronounced in the northern hemisphere, with


  more carbon is accumulating in forests and other vegetation and soils in the Northern Hemisphere during the summer, and more carbon is being released in the fall and winter


Daily as well as seasonal variations were observed in northern Japan, reported in the article Diurnal and seasonal variations in atmospheric $\mathrm{CO}_2$ in Sapporo, Japan:
Anthropogenic sources and biogenic sinks (Miyaoka et al. 2007).