It can have two meanings, I think. The main meaning I use it for is to form pairs with someone else, when you are doing some activity, usually when there is some risk. You look after your buddy, and they look after you. An example might be if you go out walking in a group. Someone might say 'let's buddy up', or 'let's all buddy up with someone'. You and the person next to you, or perhaps someone you know, become 'buddies'. You might walk together, or you might just make sure your buddy is still with the group every so often. This is usually just for the duration of that activity. It's very common, and a very important safety measure in SCUBA diving. I it can also mean to become friends with someone, e.g. 'Go on, go down to the beach. You might buddy up with someone, and then you will have someone to talk to for the rest of the week'.