There is only one reason why Please Please Me is still popular today. That reason being, its a Beatles record. Not any Beatles record. But the first in what would be a revolutionary journey that changed music forever. But, I'm getting ahead of myself.
Please Please Me isn't a bad album. It just isn't a particularly great album. It is no different from any other album from 1963. A likeable album, but one that would be forgotten not long after it was released, like many artists from that time that are completely forgotten today. The only difference from those other albums is the tag: "Made by The Beatles" and it instantly becomes legendary, a classic, an inspiration for everybody to come after it. A small, rat infested B & B in Wales can become a World Heritage Site, if The Beatles had spent a night there. The same principle works with their music.
Perhaps I am being a little harsh on Please Please Me. Songs like 'Twist & Shout', 'I Saw Her Standing There' and 'There's A Place' are good songs. Others, like 'Ask Me Why', 'P.S. I Love You', and 'Misery' just sound like songs to fill the gaps between the songs that they knew the people wanted to really hear.
Overall, by The Beatles later standards, its fairly weak. Its no Revolver, nor should it be treated like Revolver. But, it is definitely worth listening to. Consider it the first step of a brilliant musical experience from one of the most innovative bands of our time.