adobe illustrator - How do I flatten this vector logo leaving only the visible black paths?

Hey guys I've got years of experience with Photoshop but I'm new to Illustrator.

I've followed this simple tutorial to create this logo, https://www.youtube.com/watch?v=9Y9unghEExw

I want to flatten the image and make only the black lines my vector shape. Is there any way I can do this?

I've tried using the pathfinder but the entire shape goes black, possibly because its reading the White as part of the entire shape? Please advise!