The primary draw or lure for politicians to become lobbyists is, of course, money. The financial opportunities for people who have these kinds of coveted and powerful relationships are virtually limitless. They are connected not only to the politicians (their former colleagues and peers), but they also know "the people" who know "the people" who wield the power. Knowledge and relationship--as well as a cache of political favors--make a former politician the perfect candidate to be a lobbyist. This system benefits all of "them," of course; what this does for us is skew the political system in favor of those who can buy power. It's out of balance, but the only people who can change it are those who have the most to gain from keeping the status quo. Catch-22.
There is a very close relationship between those in Congress and those in business in this country (as in many countries) because Congressmen hold the checkbook for the nation. Government contracts account for hundreds of billions in spending each year, and the lobbyists are often chasing that money, or chasing regulations that might keep their employers from making more profits.
I also tend to believe that our laws against Congressmen becoming lobbyists aren't nearly strict enough. I believe there should be a ten year moratorium after a Congressman leaves office on taking a job with firms that benefited from legislation they have sponsored or written.
To me, the main reason for this is the fact that this is the easiest way for ex-lawmakers to use their experience to make a whole bunch of money.
After lawmakers leave Congress, they tend to want to make money. Their experiences in Congress do not qualify them for a whole lot of high-paying jobs, though. Their main value is in A) the fact that they understand the process of law-making that goes on in Congress and B) the fact that they still have personal connections to many of the people who continue to be important in Congress.
For these reasons, they are still very valuable to people who want to affect the law-making process. They can get access to people who make decisions and they understand how to influence them. This means they are likely to be able to be good lobbyists.
For these reasons, lobbying firms tend to want to hire some of the ex-Congresspeople.