Come into the world
Definition 1 of 2
Pro Tip 1/3Moment of Birth
Specifically relates to the exact moment a new life begins outside the mother. 
At exactly midnight, she came into the world.
Pro Tip 2/3Human Focus
Used mainly for human birth, less common for animals unless they're personified. 
The kittens came into the world, each meowing for attention.
Pro Tip 3/3Metaphorical Usage
Can be used figuratively for the 'birth' of new things like nations, eras, or movements. 
When the treaty was signed, a new era came into the world.