Why? Why not?! My generation (the 60's) embraced first "Frat" Rock (The Beach Boys, The Ventures, Paul Revere & The Raiders, etc.), then British Invasion, both the 1st wave (Beatles, Rolling Stones, Kinks, Animals, Yardbirds, etc.) and 2nd (Cream, Hendrix---though American, he was perceived as British, Led Zeppelin, Traffic). Why? The 50's generation liked Jazz, Pop vocalists, the original Rock n' Roll. Why? In the early 70's singer/songwriters (Carole King, James Taylor, Elton John, many others) ruled, and Progressive Rock was hugely popular. Why? The mid-to-late 70's saw the rise of the "anti-Prog Rock" Punk movement spearheaded by, it can be argued, The Ramones (though The MC5 and The Stooges inspired them). The early 80's was the era of Corporate Rock (name your poison ;-) and, for the trendier, New Wave. Why? The late 80's were owned by the Sunset Strip hair bands/Arena Rock "acts". Why? The 90's saw Alternative (REM, all the other "College Radio" bands) become THE music to like. Why? And then, every once in a while an Artist comes along who endures for decades---Dylan, Tom Petty, Springsteen.
Pop music is largely just fashion, no different than clothes or hair styles. How then can Rap have stay popular for so long, defying the normal rules of fashion? A person or people who don't like Rap (I have a feeling that's most Audiogoners) is/are the wrong one/s to ask. One may as well ask why jeans have been fashionable for so long. Pre-50's men did not wear jeans, sporting khaki's and slacks their entire lives. Post-50's guys rarely wear anything but jeans.
Unlike the examples in the above musical timeline, Rap didn't replace anything---it is it's own genre. It may be thought of as younger peoples Blues, which it is somewhat similar to. As with Rap, Blues was originally criticized (by white America, of course, for whom it was neither created nor intended, but also by Baptist Preachers) for it's crude/vulgar lyrics, repetitive musical structure, lack of melody, and general unlistenability, just as was 50's Rock n' Roll by the Big Band generation. That generation couldn't understand how the following one could like the music it did; I know 60's guys who continue to piss and moan about how music nowadays is no good, they not being able to accept the fact that "their" music (The Beatles in particular, it seems) is no longer being made (it is, but only on a cult level), or popular the masses. Why should they expect it to be? That is no different than their WWII parents bemoaning the death of the Big Bands. We had our time---it's now "theirs"!