A contributor’s copy is simply a copy of the issue in which your work appears. Once the issue is published, they’ll send you a free copy (or several, depending upon their policy). Many publications provide at least one contributor’s copy, whether or not they also provide monetary payment.
Literary journals often “pay” in contributor’s copies. Many simply don’t have the funds to pay writers in cash. Some are operating at a loss just producing the journal. Offering contributor’s copies at least gives the writer a look at the finished product and perhaps a few copies to share with others.
Writers certainly should be compensated better. And there are journals and magazines that are able to pay more. But the vast majority can’t do this. However, payment isn’t the only incentive for publishing in these journals. The editing process can produce stronger work. And many writers publish in these journals so that their work reaches a wider audience. Editors and agents also comb through them looking for new authors, so a publication could lead to other opportunities.