There are generally said to be two broad ways in which interest groups affect Congress. These are known as the “inside game” and the “outside game.”
In the outside game, interest groups use their members and their public relations machines to influence Congress. We can see this happening, for example, on the area of gun control. Once Congress started talking about gun control, the NRA mobilized its members to put pressure on Congress. The members started contacting members of Congress and doing other things to show Congress they did not support any gun control measures. This influences members of Congress because they know that these people will vote against them if they vote for gun control.
In the inside game, interest groups directly lobby members of Congress. They talk to them in person or on the phone and try to persuade them to vote in certain ways. They make campaign contributions. They help members of Congress to write bills that contain provisions the interest group likes. In these ways, the interest groups are working directly on Congress to influence them. This is typically said to be the most consistently effective approach.