Anansi was feared and disliked because he told stories. "All the stories were Anansi's." What's so scary about a story?
Today when we think of story tellers, we probably think fondly back to Saturday morning story-time at the library. The friendly librarian would read several books to us, we'd make something cool that was about the stories, we'd sing some songs and maybe do some dances, and go home happy.
For most of history, storytelling held a very different place in society. Before the Internet, before the six o'clock news, before daily newspapers, information was mostly spread by storytellers and bards. These travelling entertainers took news from one place to another, keeping people (mostly) up to date on the news of the country.
In ancient times, the Bard, the Fool or the Troubadour usually held an honored place in the courts of kings. They were the spin doctors of their day. Yes, they were entertaining, but they were also the archivists - they remembered and repeated the history of the kingdom.
William Shakespeare was called the Bard of Avon.
These travelling story tellers were often trouble makers. Rulers kept them close at court to keep them from telling the whole story. As long as the story tellers were employed in court, they had a vested interest in keeping the ruler generally happy, and would keep their stories and songs "clean" and politically correct. There are an awful lot of stories about troubadours or bards who insisted on singing an uncomplimentary song or telling an unappealing story, and then paid the price - banishment, blinding, losing their tongue, or death.
Being a god, Anansi didn't have any of those potential repercussions. Anansi could (and did) tell any old story he wanted. And, the story stuck. ("That's a goofy looking dog! Hey, Goofy!")
What would you do if someone started freely telling the story of your life, in the most entertaining manner possible? If it was Anansi, I'd be changing my name, and moving to the most remote desert island I could find.....