How does a zygote become an embryo?

Through growth and development.

By growth we mean increase in size and also in the number of cells. An adult human, for example, has about 10 or even 100 trillion cells. The cell divisions are mitosis.

By development we mean that as the embryo grows cells become different, a process that is called, logically enough, differentiation. Cells not only become structurally different, but they acquire correspondingly different functions, too. So, to take a human again as an example, some cells are specialized for carrying oxygen around the body in the blood, some destroy bacteria that they encounter, others pump sugars out of the intestine into the blood after a meal, and so on. This division of labor makes for a highly complex and efficient organism.

So the first thing that visibly happens to a zygote is that it divides by mitosis into two cells, then they divide to form a four-celled embryo, and so on. Eventually the ball of identical-looking cells hollows and acquires an axis, and differentiation begins.

For more detail, see the links below.