I'm new to home recording and somehow I'm still curious why my music can't sound like these pro
I didn't listen to the clip, so I'll just answer your more general question above with a question or 2 of my own.
OK, first of all, if you think you're frustrated now, get ready for years of frustration. The reason these records sound pro is because they're made by pro's. Just like a pro football player, or a pro carpneter, or a pro....well you get the point, it takes years of practice, learning, making mistakes, developping skills, and experience. It's not just a matter of how many instruments are in a recording, and there's no special "technique" or "secret".
Now, to hopefully help answer your question without even listening to the clip you posted, I have a few questions that, once answered, my might help others answer your specific questions on the clip.
Where are you recording (bedroom, basement, bathroom?, etc....)
Is the room treated for recording and/or mixing?
What kind of equipment are you using? In this case what kind of mic?
What are you plugging your mic into?
It would be hard to tell you what's missing without knowing what you have and a general idea of how and where you're recording.