While SingerOfTheFall's answer seems correct, it isn't. When you find out about Xcution (organization of Fullbringers) you learn that they have been in an incident with a Hollow as a child. As in, a Hollow killed their parents when they were young.
In Ichigo's case, his mother was attacked by a Hollow, thereby giving Ichigo Hollow powers.
From the wiki on how Fullbringers are made,
However, they differ in that they each have a parent who survived a Hollow attack before they were born. As a result, traces of the Hollows’ power remained in their mothers’ bodies, which were then passed onto them at the moment of their birth.
However, it probably didn't awaken until he was pushed to the edge by "training" with Urahara due to the Shinigami powers overwhelming the Hollow powers. His situation is similar to what happened in the manga recently (major spoiler):
Ichigo got Quincy powers by releasing a massive amount of spiritual energy. He got Quincy powers because his mother was a Quincy.