General aviation pilot reported they did not follow ATC departure instructions resulting in failure to meet obstacle clearance requirements. Pilot corrected and continued on course.

Narrative

I took off IFR from ZZZ Runway XXL. Was given runway heading and my climb altitude was 7;000 ft. MSL. After takeoff; tower handed me off to Denver Center for the start of my trip back to ZZZ1. After making contact; I understood Denver Center had cleared me direct to GCK in my climb and I read back that I was proceeding to 7;000 ft. and direct to GCK. After a period of less than a minute; Denver Center questioned why I had turned eastbound direct to GCK. They advised that I had been cleared to GCK AFTER reaching 7;000 feet in order to meet obstacle clearance requirements for the departure. I immediately turned back to my original heading and advised ATC of this. I apologized to them for the misunderstanding and the center controller advised that it was not a problem. Upon reaching 7;000 feet; I asked Denver Center if I could return on-course direct to GCK and they acknowledged this would be acceptable. The main problem was that based on my misunderstanding of the original clearance; I did not follow the ATC instruction to not continue on course to GCK until first reaching 7;000 feet.