The information to make proteins is carried in DNA, in a unit called a gene. When it's time to make a protein, the DNA gets unzipped and a polymerase reads it, making a complementary copy of it in RNA. That RNA goes to another molecular machine, the ribosome, which reads it and translates the RNA code into a protein. But sometimes, there's another step in there: splicing.
Some genes carry extra information in their DNA. There will be a stretch of useful information (an exon), and then a long stretch of garbage (an intron), then another stretch of good stuff, then some useless crap, then some good stuff again. The whole thing gets transcribed into RNA, so the RNA contains the garbage too. Before the RNA goes to the ribosome, the garbage gets cut out and thrown away, and the useful bits get spliced together. (There are sequences in the code that specify what's crap and what's not.) Usually, proteins do the splicing, but in some cases, the garbage RNA intron splices itself out, without the use of proteins.
The splicing reaction happens in two steps. First, something in the middle of the garbage segment, a particular purine, attacks one end where the good meets up with the useless, separating them and binding to the useless end. (This forms a loop.) Then the loose end of the good stuff attacks the next segment of good, breaking it away from the looped garbage.
This reaction involves breaking and making bonds in the RNA polymer, just the kind of job we thought only a protein could do. In most splicing reactions, proteins and other RNAs assemble the pieces in the right position for the reaction to occur. But the reaction itself happens on its own. Even the self-splicing introns, which use different molecules to achieve virtually the same result, simply put the players in position, and let them do their thing.
When we thought only a polymerase protein could break and make bonds, it was naive, because even polymerases are simply organizing the molecules and then watching the reaction happen.
So we now have evidence that RNA can make and break bonds, simply by positioning the right things in the right place. But can it really self-replicate?