The Bilderbergers are attendees of an annual meeting between the planet's top leaders in the corporate, banking, and political world. The inner core & steering committee is driven by those involved with the international central banking system, and their steady membership also breaks off into other thinktanks and lobbying circles such as the Trilateral Commission, Club of Rome, Council on Foreign Relations, and Royal Institute for International Affairs.
They aren't mentioned in the mainstream news because they own most of the mainstream news.
Yes, they are very influential. It is a proven fact of history that they planned for the slow evolution of European free trade agreements into a continental superstate, which has come to fruition with the EU. They have made similar lobbies for North America, which have led to NAFTA, SPP, and other transnational agreements. Scholars and experts on the subject have deduced from documents and publications by their spiderweb of subsidiary thinktanks that they eventually plan to build a world superstate with a global parliament, global central bank, courts, and army/security force. Expressions of this are already seen in the direct involvement of their members in the UN, IMF, World Bank, NATO, etc.