Cessna 172 pilot reported while taxiing onto the runway for departure they failed to observe an aircraft on short final. The aircraft on short final executed a go-around to avoid any further conflict.

Narrative

I was second in line to depart Runway XX at ZZZ. It was about XA:40 p.m. and the pattern was very busy with many aircraft arriving; departing; doing pattern work; and shooting approaches. I was not prepared for that much traffic and lost track of where all the aircraft were in the pattern. I saw one or two aircraft land or shoot touch-and-goes before the Piper in front of me entered the runway. After the Piper began his takeoff roll; I pulled up to the hold-short line; positioning my aircraft exactly perpendicular to the hold-short line so I could view its departure. I heard an aircraft announce five miles out from entering a left downwind for XX and as soon as the Piper was off the ground; I announced I was entering XX for immediate takeoff. I failed to check final again. As soon as I crossed the hold-short line; an aircraft said; 'Aircraft entering runway XX; stop.' I stopped immediately - before leaving the taxiway - looked left and saw a twin engine aircraft with all its lights illuminated on short final at perhaps 150-200 feet. Stunned; I replied; 'Aircraft X holding short.' The landing aircraft said; 'You are not holding short! You've entered the runway!' He gave his call sign and announced he was going around.My aircraft was fully across the hold-short line; but there were probably 10-20 feet of the taxiway ahead of me when I stopped.The other aircraft did not deviate from being over the pavement; and could have landed safely; but the pilot did the right thing to go around. I estimate we were within 150 feet of each other horizontally and vertically at the closet point. He flew the runway heading and as soon as he began his climb out; I checked final again; announced my takeoff; entered the runway and departed without further incident.In retrospect; when pulling up to the hold-short line; I should have not have turned perpendicular and should have allowed myself a better view of the full final approach; because my left wing obscured the view of the middle part of final approach. I could only see where aircraft were turning from downwind to base; then base to final; and then the last several hundred feet of final. I let myself get in a hurry to depart and failed to check final again before crossing the hold-short line. My passenger; a pilot I was picking up at ZZZ; said he never saw the traffic either; but I'm not sure he was looking as I should have been.I was pretty much in shock on the return flight to ZZZ1 that I had been involved in a runway incursion. I pride myself on having a safety-first attitude; but I had a major lapse today. I was distracted by all the traffic and radio calls; was more focused on my departure than on arriving traffic; and lost situational awareness the twin's position in the pattern. I plan to discuss this at length with my instructor.