tl;dr: Select your layer, grab your pen tool (G), and click and drag on the little keyframe square in the Composition window. This will drag out bezier handles that you can use to fine tune your motion path curve. Long answer: I believe I had a different setting than default enabled while recording this tutorial. The quick solution is to use the pen tool to fine tune your motion path's bezier handles in the Composition window, like in the tl;dr above. If you want it to behave like you see in the tutorial, select all of your diamond keyframes in the timeline, right click, select "Keyframe Interpolation", and change "Spatial Interpolation" to "Bezier".

@ thank you.

I think the music is fine, but the volume is too loud. Makes it hard to hear what's being said.