dayspring is a compound word, composed of day and spring.
day in this case is our common understanding of that portion of a 24-hour period in which there is sunlight.
spring is used in an archaic sense of "origin", "beginning", or "source". (It's actually from a Germanic word that means an upwelling of water, such as a well or stream.)
From a more technical perspective, "sunrise" is an approximate synonym. I highly encourage you to watch a sunrise, to get a sense of the changes that occur from the period of complete darkness until the sun is fully visible.
When we were dating, my (now) wife and I watched a sunrise from a boardwalk on the Atlantic Ocean coast of the United States. We arrived about a half hour prior to sunrise. It was quite dark, except for the street lamps behind us. Ahead was pitch black with the sound of ocean waves breaking.
After a few moments, the horizon began to reveal itself - the boundary between sea and sky. While the sea remained relatively dark, the sky shifted through shades of gray, blue, yellow, and then white. Streaks (rays*) of light reflected through the wispy clouds. Eventually, a deep orange sliver of the sun peeked above the horizon. Along with the sea and the land, we were immediately bathed in light. The light grew and grew as the sun climbed, until it was fully revealed.
In my mind, then, the sunrise is that entire process, while dayspring consists of that instant where light suddenly breaks across everything around you.
* Regarding light rays, these are actually formed when clouds partially block sunlight, at different levels of dispersion. In some places the light will be more completely blocked, while in others it streams through relatively unhindered. This results in a "ray" or beam effect, even though from a purely physical perspective it's somewhat of an illusion.