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The basic mechanism
The Earth receives energy from the Sun in the form of radiation. To the extent that the Earth is in a steady state, the energy stored in the atmosphere and ocean does not change in time, so energy equal to the incident solar radiation must be radiated back to space. Radiation leaving the Earth takes two forms: reflected solar radiation and emitted thermal infrared radiation. The Earth reflects about 30% of the incident solar flux; the remaining 70% is absorbed, warms the land, atmosphere and oceans, and powers life on this planet. Eventually this energy is reradiated to space as infrared photons. This thermal, infrared radiation increases with increasing temperature. One can think of the Earth's temperature as being determined by the requirement that it produce the infrared flux needed to balance the absorbed solar flux.

Solar radiation at top of atmosphere and at Earth's surface.
The key to the greenhouse effect is the fact that the atmosphere is relatively transparent to visible solar radiation but strongly absorbing at the wavelengths of the thermal infrared radiation emitted by the surface and the atmosphere. The visible solar radiation heats the surface, not the atmosphere. Whereas most of the infrared radiation escaping to space is being emitted from the upper atmosphere, not the surface. The infrared photons emitted by the surface are mostly absorbed by the atmosphere and do not escape directly to space.

Atmospheric transmittance of various wavelengths of electromagnetic radiation (measured along sea level).
The reason why this results in a warming of the surface is most easily understood by starting with a model of a purely radiative greenhouse effect, in which one ignores the fact that a large part of the energy transfer in the atmosphere is not in fact radiative, but associated with 1) convection, (sensible heat transport), and 2) the evaporation and condensation of water vapor, or latent heat transport. In this purely radiative case, one can think of the atmosphere as emitting infrared radiation both upwards and downwards. The upward infrared flux emitted by the surface must balance not only the absorbed solar flux but also this downward infrared flux emitted by the atmosphere. The surface temperature must rise until the surface generates enough thermal radiation to balance the sum of these two incident radiation streams. 

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