To add to Viper1's reply:
Most older content - games, television, movies - displays in 480i only, even if you display it on a TV capable of a better resolution. Some games last gen were capable of 480p, and pretty much everything on the PS3 and 360 is capable of displaying in at least 720p. If you display a 720p game on a 480i television, however, it will also only display in 480i.
Therefore...
480i content + 720p/1080p display = 480i output
720p/1080p content + 480i display = 480i output
720p/1080p content + 720p/1080p display = 720p/1080p output
In other words, both the content (the source media) and the display device (your TV) need to be capable of higher resolutions for it to display in HD and/or progressive scan.
Keep in mind, too, that many newer DVD players can "upscale" standard 480i DVDs to display in 720p/1080p. You see the same phenomenon in the X360 and PS3's software backwards compatibility, which can upscale old Xbox and PS2 games, respectively, in the same manner.
Also, to answer your above question: Games that render in 640x480 (or the widescreen equivalent) can only draw lines or textures for that many pixels on screen. 720p and 1080p allow for many more pixels on screen at once, so that games are capable of much more detailed textures and less of the "jaggies" you saw in last-gen games.
"'Casual games' are something the 'Game Industry' invented to explain away the Wii success instead of actually listening or looking at what Nintendo did. There is no 'casual strategy' from Nintendo. 'Accessible strategy', yes, but ‘casual gamers’ is just the 'Game Industry''s polite way of saying what they feel: 'retarded gamers'."
-Sean Malstrom