The plot shows the land-ocean temperatures for the years that saw a huge increase of the quantity of carbon that humanity has added to the atmosphere. The points are yearly values and the red curve a five-year average, which smooths out the fluctuations to highlight the general trend (source: NASA with data at https://data.giss.nasa.gov/gistemp/graphs_v3/Fig.A2.txt). The shaded area shows the total carbon emission each year from human activities (source: Department of Energy with data at http://cdiac.ornl.gov/ftp/ndp030/global.1751_2011.ems).
I won't go into details of how heating of the oceans and air affect climate because this phenomenon is well known and not a topic of debate. The point of contention is proof that humans are responsible. The plot shows excellent correlation between the mass of greenhouse gasses that are emitted by human activities and the temperature. But correlation is not proof of causation, so how do we know that we are to blame?
The science is too complex to argue in a public forum. The reason for the high degree of certainty is based on the many different streams of evidence that converge on the same conclusion. It's like all compasses pointing to magnetic north; those that don't are found to be broken and point randomly, refuting the assertion that the vast number of north-pointing compasses must be wrong.
This convergence of many independent lines of evidence in support of a conclusion is called consilience, a topic that is well explained by Michael Shermer in Scientific American, and can be found at https://www.scientificamerican.com/article/why-climate-skeptics-are-wrong/. This is a must read for all non-scientists who want to understand how science works and why it is so reliable.
Some people may blame scientists, who base professional judgements on extensive knowledge and consilience, of being arrogant. A proclamation of a "truth" that goes against the accumulated evidence of a scientific field is true arrogance. Basing policy that goes against consilience is sheer folly and muzzling the fruits of consilience is the ultimate crime.
It would be a waste of my time to fully analyze the literature on my own. I trust the scientists working in the field who build high-tech apparatuses, take data over decades, and meticulously analyze it. After all, the consilience they have reached is the result of extensive professional debate. It is arrogant for citizens without expertise in a field to deny anthropomorphic climate change based on the fact that Moscow Idaho has been exceptionally cold this winter. Policy should be based on the best evidence that science has to offer, and the scientists -- as a group -- with expertise in areas related to climate science are the ones who should be trusted by the public.
NOTE:
Global warming is a scientifically imprecise term because it focuses on the temperature increase, which is a consequence of extra trapped energy. When light from the sun is absorbed by air, the earth's surface, and oceans, that energy is converted to heat. Heat is simply the flow of energy in the form of motions of atoms and molecules. Some of the energy of the warm earth is radiated through infrared light, but greenhouse gasses absorb that light, preventing it from escaping the earth's atmosphere, thus converting it to heat. Greenhouse gasses pass sunlight but block the infrared light that is needed for cooling, leading to a buildup of heat and thus an increase of the temperature.
The temperature increase due to a unit of heat energy depends on the mass of the object. Since the oceans are so much more massive than the earth's atmosphere, the average ocean temperature will increase less than the atmosphere for a given amount of heating. Since the energy imbalance driven by greenhouse gases is the key process at work, ocean and atmosphere temperatures must be combined if the effects of greenhouse gasses are to be accurately assessed. That is why the land-ocean temperature composite is a good index of global heating.
