Where do babies come from? Storks, of course!
In order to, erm, grow the population from the, um, inside out, you need to place a male and a female into living quarters. Charisma has a massive effect on this, particularly the time it takes for the two to decide to get together. First they talk, then they dance, then they run into the back of the room... he runs out happy, she limps out, holding her back and wearing a yellow shirt. She is now pregnant.
You'll need to upgrade your living quarters, now.
A key to making this faster is to have both male and female max out their charisma. Have each wear clothing that increases charisma, and it'll go faster.
Pregnant women still use any gear you give them, so any clothing perks they have will still apply. This is important in maximizing efficiency. However, you'll want to consider taking good weapons away from them while they're pregnant. Pregnant women run away from fights.
After several hours, you'll be told that a baby is coming. Click on the little green baby head icon, and the baby will then join the population, and she will revert to whatever gear she is wearing. If you are working to grow the population quickly, you can again place her in quarters with a man, and the circle of life repeats.
Of note: babies will not be born when your vault is full (they stay pregnant), so either dwellers will need to die or be exiled first.