"But what do I say?" The actual process of networking can seem somewhat mysterious, yet it really isn't. You are simply making contact with individuals who may be able to assist in your job search.
The best method for making contact with others is to simply ask for advice. You are not asking for an interview, you are asking for advice. By making advice requests, you can tap into a vast network of people who can assist you in your job search.
The Advice Request Technique is the door opener to asking questions of your network contacts. You can preface your question with: "May I ask your advice about something?" Then ask away.
Follow up your advice request with a brief description of your background and what you are seeking, then use this question: "Which are the top five employers that you recommend I contact in my job search?" Most people are more than willing to help you. It plays well on the vanity factor of who they know and opens the lines of communication for you to ask further questions. Many are able to give a "Top Five" (or at least a "Top Three") list quite easily.
After they give you the names, ask them if they have any personal contacts at any of those companies. Keep in mind that over time you will begin to hear some of the same company names being repeated (especially if they are well-known industry leaders). But as you keep asking you will also gain new personal contacts—and you can never have enough. After they give you the contact names, selectively consider (depending on your comfort level with the individual) asking if they will reach out to the employer contact on your behalf. Sometimes your network contact can do some of the work for you! This is especially effective when the contact is within their company, since it can constitute an employee referral. Yet it is a common professional courtesy to connect contacts from your network to one another by way of an introduction.
When they make the referrals, be certain that you follow through in a professional manner since they are putting their personal/professional reputation on the line for you. Attaching their name with your name puts a heavy responsibility on you to meet or exceed all expectations.