How do we know that the warming of the past century is not due to the Sun?

Two compelling arguments rule this out:

1. Scientists have been observing solar brightness for several decades now, and we see no trend in the Sun's activity that correlates with the rise in global temperature.

2. If the Sun were causing recent warming, the warming would occur throughout the Earth's atmosphere – in the upper part (stratosphere) as well as the lower part (troposphere). In fact, the stratosphere is cooling as the troposphere is warming. This cooling results partly from stratospheric ozone depletion and partly from the fact that as greenhouse gases in the troposphere trap energy, less is transmitted into the stratosphere. This observation supports the greenhouse-gas explanation for recent warming.

How do we know that the warming of the past century is not due to changes in volcanic activity?

Explosive volcanoes act to cool climate because they place reflective particles and droplets into the stratosphere. So we'd need to have a significant reduction in explosive volcanic activity in order to explain warming. No such reduction has been documented. In fact we've had two large explosive eruptions since warming intensified in the late 20^th century, and fewer in the mid-20^th century before the rapid rise in temperature. (Volcanoes need to be highly explosive, ejecting material high into the stratosphere, in order to influence climate; oozing lavas in Iceland and Hawaii don’t count.)