The blind spot is the spot on your retina where the optic nerve fibers connect and exit out of the eye. It's the only part of the retina that doesn't have rods or cones.
There are two major theories about why we are not aware of our blind spot when both our eyes are open. One theory suggests that the blind spot of each eye falls into the line of vision of the other eye. According to this theory, each retina compensates for the blind spot of the other.
The more widely accepted theory is that the brain "fills in" missing information from our blind spots. The exact mechanism of how this works is unknown.
Neuroscientists do know that all visual information is constructed in the brain. The eye doesn't send visual images to the brain. Rather, the brain constructs visual images based on chemical feedback from rods and cones. Rods and cones are photoreceptors: they "light up" differently when different wavelengths of light hit them. The brain uses this information to construct what is happening visually. Since all visual construction occurs in the brain (not the eye), it makes sense that the brain would be able to complete the picture, despite blind spots.