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You have to go into the settings and set it to record and save to your computer. There are different versions of OBS, but on mine you go to settings then pick "broadcast settings", and check off "automatically save stream to file" and pick where you want it to go.

Heh, I was going to link you to the channel where I uploaded my videos, but I just realized how much I was always cursing and it doesn't seem appropriate at the moment.