What is the nature of this organization that seems to give them influence?
There was obviously a need for such a forum, and preferably one not quite as scandalous as the Bohemian Grove meeting. If there was no such need the WEF wouldn't exist. The Bohemian Grove meeting could have met that need, but it is also associated with a bit too much secrecy and a bit too much scandal. The Bohemian Grove meeting also is a men's only meeting.
In addition to obviously making a lot of money for the WEF, making the price of admission high keeps the riffraff out (or at least on the sidelines), and that is part of the attraction to both businesspeople and to politicians.
The wikipedia page calls the WEF a lobbyist organization. That's not quite accurate. Certainly lobbyists do attend, but some are at odds with one another. But so do high level businesspeople attend, once again some at odds with one another. Finally, so do politicians.
The WEF is a forum for people with enough money (attendance is very expensive) to attend, whether that admission price from the businesses they own, from the donations they receive, or from their own pockets. I've been to conferences where my employer groused about paying a $1000 attendance fee. The WEF costs 120 times that.
That high admission price is very intentional. It keeps the riffraff out. Or rather, it keeps the riffraff outside the conference doors. That a lot of high level people do attend draws a lot of press and a lot of protesters who have to stay outside the conference doors.
That an ever increasing number of people do attend makes the conference ever more attractive to high level people who can afford to attend. The G7 and G20 meetings are government only. The United Nations is government only. The WEF meeting is open to government and to business. There's nothing else like it.
They seem to produce opinions and speeches.
The important discussions almost certainly happen in hallways and in side rooms. That's where the important discussions happen in almost every conference.