Why does Beethoven spend all his time erasing music?

He didn't spend ALL of his time erasing music. If that were true, we wouldn't have any music of his to enjoy. He did write a lot of sketches while he composed, and he edited and re-wrote a lot. The legend and possibly the myth around Beethoven is that he did this to an extreme degree. I'm not sure whether or not this is true. This legend or myth may have developed in order to contrast him with Mozart, for whom a quite different legend arose. Many believe that Mozart virtually never edited or changed his music, and that he created entire compositions in his head; it only remained for him to jot down the notes when it was convenient. Both of these legends may be a little less true than most people assume. What may be true is that Beethoven had to work and re-work material until he found the right stuff, and Mozart may have been a little more facile than Beethoven. I personally find Beethoven more appealing because he had to (and did) apparently exert more effort in order to produce his works of genius.

Additional: Beethoven didn't erase anything. He crossed out a lot, usually with bold, angry-looking strokes. He probably tore some up and threw it away, but we have most of his notebooks and scores.