Interesting question! First, just to be clear, not all babies are born with blue eyes. But, many are indeed born with blueish eyes that then change color.

You can learn quite a bit about the genetics and biology of eye color (it's complicated!) on a wiki site that is written and maintained by scientists who study this:

In human eyes, variations in color are due to varying ratios of a pigment called eumelanin produced by melanocytes - a special type of cell - in the iris of the eye. The simplistic answer to your question is that the pigment in the eyes has not accumulated enough yet to give a darker color.

In European populations, children are most commonly born with unpigmented (blueish/grayish) eyes. As the child ages, melanocytes slowly begin to produce melanin. Because melanocyte cells continually produce pigment, eye color can change (usually darkening) and this typically happens in the first year.