You can record your streaming sessions for posterity's sake -- either to watch yourself (to see what you're doing right/wrong), or to post to video sites like YouTube for your followers to view.
Here's how to record videos in OBS:
Set your recording folder
To set up where your recordings go, go to Settings > Broadcast Settings. Next to File Path, click Browse...
Find the folder you want to save your recordings in. You will need to enter a file name -- you can use character sequences to automatically label your recordings by date, time, etc.
So, if you type $Y$0M$0D in the file name box, your videos will be saved as 20160128 and so forth.
You can also choose what type of video file you want OBS recordings to be saved as -- .flv or .mp4 -- by selecting the file you want from the file type drop-down menu.
Record automatically whenever you go live
If you want OBS to automatically record your stream whenever you go live, go to Settings > Broadcast Settings and check the box next to Automatically save stream to file.
Record a test stream
If you want to test what your stream looks like before you go live, you can record it (without going live) and play it back to see what you need to change. To do this, simply click Start Recording from OBS' main window.
Post your past broadcasts on Twitch
You actually don't need to record your streams in OBS to use Twitch's Past Broadcasts feature. Twitch will automatically archive your past broadcasts for you, though they will not remain on the site indefinitely.
To set this up, open Twitch and go to Settings > Channel & Videos. Under Channel Settings, check the box next to Archive Broadcasts.
Your broadcasts will now be archived on Twitch -- but not forever. If you want to save your past sessions forever, you'll need to record them in OBS.