For attribution and quantification of anthropogenic warming, we rely on physics and math. GHG mixing ratios are accurately monitored. Using lab spectroscopy, we can empirically determine the radiative forcing change resulting from a change in GHG concentration. These lab-derived calculations (based on 150 years of physics) are corroborated by empirical measurements of actual changes in radiative forcing over time derived from satellite-based radiometers. We know how much forcing change to expect from a known release of carbon (ΔF = 5.35 ln (C/C₀)). We know how much carbon we have released, and we additionally have the ability to trace that carbon (through isotopic fingerprinting) to fossil fuel burning. We can then assign a range of expected temperature change based on estimations of climate sensitivity (the amount of mitigation of forcing by feedbacks).

Here is a summary of forcing changes relative to the year 1750:
http://www.realclimate.org/images/ipcc_rad_forc_ar5.jpg