How is color information transmitted from the eye to the brain via the optic nerve?
Color vision in layman's terms: 

The eyes contain a photosensitive layer much like a digital camera called the retina. In the center of this layer is a 3 mm-wide spot called the fovea that contains high densities of photosensitive cells, called the cones. Cones are photon hungry and operate only in bright light (photopic vision). They are called cones because of their conical shape. Rods are abundant in the periphery and mediate gray-scale night vision (scotopic vision).

The cones in the fovea are present in three types: red, green and blue cones - hence they convey color images much like digital pictures, namely as a bunch of pixels with a mixture of red, green and blue. 

The cones in the retina are the first nerve cells (called neurons) in a long chain of neurons. The cones are hence called primary neurons. They connect to secondary neurons, namely the retinal ganglion cells (RGCs). These cells have long protrusions called axons. These axons bundle together to form the optic nerve. Each cone in the fovea connects to one RGC. The optic nerve extends from the retina to the brain. Information is sent through the optic nerve in the form of electrical pulses, just like a photosensitive chip operates in a camera. The information from the optic nerve eventually reaches the primary visual cortex in the brain, also referred to as V1 or striate cortex (Fig. 1).  

In the visual cortex, there is a retinotopic map, meaning that a topographic representation exists in the brain that matches the retinal image. In other words, images are projected faithfully on the brain's surface as neuronal activity. The brain's job is to process these images further and generate conscious perceptions out of them; we see with the brain, not the eyes (a famous quote from Bach-y-Rita).   


Fig. 1. The visual system. source: IEEE

References
- Kolb et al., Webvision. The Organization of the Retina and Visual System. Utah University
- Purves et al., Neuroscience 2nd ed. Sunderland (MA): Sinauer Associates; 2001