It means to begin a friendship, or a conversation. This phrasal verb can't be separated.

eg：We struck up a friendship on the first day we met.
    While he was waiting in the queue, he struck up a conversation
    with the person in front of him.

    Note:

    strike up的意思为发起，但多数时候用于发起谈话。strike和up不能分开。

    例：我们第一次见面就成为了朋友。

    排队的时候，他开始和前面的人聊天。