There are a number of reasons for this.
First, we can say interest groups have power because they are able to donate large amounts of money to political leaders in one way or another. Officials need that money to fund their campaigns and the interest groups are often able to give it. This allows interest groups to (at the very least) have their voices heard by officials to whom they contribute.
Second, laws and policy issues are very complex and interest groups have experts who understand them. Members of Congress often cannot understand the complexities of all the various issues that face them. Interest groups have experts who can help the Congresspeople understand the issues or at least know how to act on the issues. This saves legislators the time and effort needed to learn about things on their own. This allows the interest groups to influence the minds of members of Congress as they educate those members.
Finally, interest groups have power because they have the backing of groups of people. Interest groups generally represent large numbers of people like gun owners or senior citizens or business owners. These interest groups can help to mobilize those groups to participate politically. The fact that they have votes behind them helps make interest groups powerful.
All three of these factors help to give interest groups power in the political process.