1) Temp Visitors
You could tell your visitors to read a sample issue of your free e-zine on your web site. For example, you could say, "Read a sample issue before you subscribe to our free e-zine!" Another example, "Check out a sample article before you decide to subscribe to our free e-zine!"
2) Bribe Visitors
You could offer your visitors a free e-book if they subscribe to your free e-zine. For example, you could say, "Subscribe to our free e-zine and get a our new e-book for free!" Another example, "Subscribe to our free e-zine and get five e-books with full give-away rights!"
3) Convince Visitors
You could publish some of your e-zine subscriber's testimonials on your web site. For example, you could say, "Why put off subscribing? Just see what other subscribers are saying!" Another example, "Subscribe today and experience the benefits the people below are experiencing!"