Basically, you've got to get on the radar of other Linkedin users and prove that you are a genuine person with expertise in your field. A couple of really good ways of doing this are using Linkedin Groups and Answers.
There are many groups on the site and they are quite active. While some of them have been a bit overrun by spammers most haven't. When participating in them it's really important to have a good photo of yourself in your profile. This appears as an avatar next to your contributions, comments and likes. If your face keeps popping up in these discussions, then it's inevitable that some people will have a look at your profile to learn more about you. Needless to say, the odds of this go way up if what you're adding is thoughtful, relevant and interesting. Keep doing this and some of those who have arrived at your profile this way will decide to send you an invitation.
As you browse the many posts and discussions in your groups, you'll identity certain people that you want to connect with. You can keep an eye on their activity in these discussions by "following" them. If you "like" their contributions from time to time and comment on them you will certainly get on their radar. (Of course you shouldn't be obsequious when doing this. That might just put them off.)
After a while you can then send them a connection invite. They'll know who you are, and the chances that they'll accept are far greater than if you did it without any prior contact.
Another good option is to use the Answers section, in which people seek the advice of countless professionals. It's definitely worth scouring this section for questions related to your interests. Offering thoughtful solutions to the queries listed is an excellent way of showing your expertise at the same time as lifting your profile. Remember that your answers will be seen by other answerers and browsers, as well as the original questioner. And if you write what he judges to be the best answer then that will be recorded in your profile, increasing your credibility.
You can also get great benefit simply by asking questions. After you get some responses you should say thanks to each of those who were kind enough to answer. That's another way of getting on their radar personally. (This is actually a surprisingly effective method. I've asked three questions so far, and thanked everyone. As a result I've had two invitations which I've accepted.)
UPDATE: Linkedin Answers has just been removed from the site! Still, the groups remain, thankfully.