Lobbyists, or people who represent special interest groups, are paid to go to legislators and convince them to support their side. Nowadays it can be done through e-mail, calls, and other forms of virtual communications, but back in the day, lobbyists would make their arguments by physically visiting legislators in the lobby of their building.
They would wait there until they found a legislator they wanted to convince and would argue their side for the special interest groups. Because they would wait in these lobbies for the legislators, they were given the name of “lobbyists.”