The first generation of HDTVs, circa 2000, were rear projection sets (CRTs) and 1080i actually came before 720p because of 720p's higher pixel clock requiring faster electronics. I have one of these behemoths in the living room -- it is 55" and weighs about 240 lbs -- and it actually looks fine, great even, at 1080i in the same way that regular televisions look fine at 480i. The interlacement just isn't an issue with CRT technology due to phosphor persistence and the other mechanics involved. However an LCD can't physically display 60 unique interlaced fields per second like a CRT, they have to process it (read: deinterlace) in order to derive and display a progressive frame. There are many different deinterlacing algorithms and processors, but most LCDs under $1500 just don't look good with interlaced content (ghosting, flickering, etc), which is why 1080p is far more preferable for displaying on an LCD.
McFly didn't say whether he was running at 1080p or 1080i, only that he was running at 1920x1080. Although since he is running an NVIDIA card in all likelihood he is running at 1080i. NVIDIA for whatever reason refuses to output 1080p over component even if the display is capable of it (I know mine is because my Xbox 360 can do 1080p over component just fine).