I believe they disliked him - mostly - because he was a ham, he hogged the spotlight, he demanded all the attention be put on him, he actively sought to cut others' lines so as to give himself more dialogue, that kind of thing.
There was a recent History Channel special in the last year and it kinds of hints at his ego issues. It was about Star Trek technology becoming a reality and Bill Shatner was the host. And for the commercial plugs, it was him saying,
"It was me me me - I did all these things..."
I'm sure he's mellowed because he's not the draw he once was. I do congratulate him, though. Somehow, he's made himself a 2nd career late in life as a writer, on TV and films portraying himself and as a character actor and I'm sure that also chapped some asses even worse.
One of the loveable things about Shatner is he parodies himself. In many ways the dislike of Shatner by some of the other TOS actors is a reflection on themselves, everyone knows that Shatner is a ham, so what? He made that show a success along with Nimoy.
I especially see a lot of brittle "sour grapes" in Takei, he impresses me as someone who is as full of himself as Shatner, with none of Shatners' sense of humor and rogueish charm.