The Becoming refers to the awakening of powers in a witch. In a Witch, magic is linked to emotions, and this link usually appears around ages 6-7. One heritage implies they can appear earlier.
13 Magazine eventually clarifies on page 18 that some girls' magical powers can appear as early as 2-3, though this is rare. An image on page 70 of the main book also shows a very small baby using magic.
Young Witches discover that when they feel an intense emotion, things related to that emotion happen. Anger may cause a glass to shatter; glee, butterflies to spawn; and sadness, clouds overhead to darken. It's at this time (called the Becoming) that Witches are usually told about their true natures.
This is elaborated on in page 28 of Adventures.
- Nightbane, an RPG which also uses the term 'becoming' for when its characters gain supernatural abilities.