This phrase can have more than one meaning, but perhaps the most common meaning is to make an offer of contact or friendship to someone. So if I say, "My sister and I have not spoken in a long time, but I am going to reach out to her this week," then I will call her or send a letter to try and make contact or open up a dialogue with her. If someone is new at my school, I might "reach out" to them- meaning that I would talk to them or something in order to become their friend.
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