Start the buzz. A reason why you may be offered advanced reading copies is your stated willingness to promote the book by telling others what you thought about it. Promoting a book you read from an advanced reading copy plays a role in whether you will receive even more advanced reading copies.
Ask for advanced reading copies. Maybe you want the latest release from your favorite author, or maybe you wish to become a regular reviewer, eligible to get on publisher reviewers' lists. You can go to your local bookstore and ask whether they've received any advanced reading copies or galleys. If so, ask whether you can read them, promising to review them for the bookstore. Be sure to send a review to the publisher as well. This alone may get you onto a reviewer list.
Another way to receive advanced reading copies is to contact an author and ask to be an "influencer" for his book. You may also go online to contact the publisher. Many larger publishers have forms to fill out or an e-mail address to use for those with book review requests for advanced reading copies.
Review the books you read. Advanced reading copies cost the publisher money. Publishers look for legitimate book reviewers or those who wish to review books---not just someone who gets a thrill out of reading a book before anyone else. Reviewing a book means giving a teaser about the book without giving away major plot points, adding how you feel about the book and a blurb the publisher can use to promote the book. Be sure to send a copy of or a link to the review to the editor who sent the advanced reading copy. (This means keeping records of books received.) If you do not have your own review site or blog, post the review online at such sites as Amazon or Barnes & Noble. If you are consistent, write good reviews and promote well, publishers will be happy to send you advanced reading copies on a regular basis.
Follow the rules and do not sell advanced reading copies. Part of reviewing advanced reading copies has to do with integrity. If the editors consistently see advanced reading copies of their books on sale at places such as eBay, they will know reviewers are doing what they are asked not to do---selling the advanced reading copies. Doing so decreases the value of the actual published book. But you can get advanced reading copies if you are willing to ask, willing to review online and willing to agree to publisher stipulations about handling the advanced reading copies.