Well let’s see, here’s a few facts that are best explained by the hypothesis that there was once some sort of RNA world, and that the translation system and genetic code evolved in this RNA world:
First of all, logically speaking, in so far as there has to be some solution to the chicken-and-egg paradox concerning the relationship between DNA as a relatively inert information storage molecule, and proteins as the so-called workhorses of biochemistry, the fact alone that RNA is capable of serving both information storage and catalytic roles lends itself as an obvious candidate. Historically speaking it’s important to mention that the RNA world hypothesis was postulated even before the catalytic roles of RNA in translation were known about, and was based only on the observation that messenger RNA servers as an intermediate storage molecule during translation (at the time it was thought proteins decode the mRNA and catalyze peptide bond formation)
But not only that, the fact that it performs that very role in protein biosynthesis can’t just be ignored.
The ribosome being a universally conserved ribozyme, RNA being an intermediate in translation, the role of tRNA in the same(oh look, RNA provides the direct structural and physical link between nucleic acid sequence and amino acid too), and that all the individual steps necessary for translation can be performed by RNA alone would seem an absurd coincidence,that RNA is prior to DNA in the pathways of nucleotide biosynthesis(biosynthesis of deoxyribonucleotides is by reduction of ribonucleotides, implying that DNA biosynthesis is an evolutionary elaboration on the pathways to RNA biosynthesis, and DNA therefore came chronologically later), and the widespread use of nucleotide cofactors in enzymes functioning in central metabolism is all data best explained by there having once been an RNA world.
We don’t have to know how that RNA world arose to be able to say that there really is evidence it existed , and that extant life appears to have evolved from some RNA-world like state, even if many aspects of the nature of this RNA world are still unknown.
There are details about how the ribosome works that only the RNA world hypothesis explains. The ribosome has three functions it performs during translation. It catalyzes peptide bond formation. It shuttles tRNA(still not physically necessary that the structural bridge between nucleic acid and amino acid should be made of RNA) between the small and large ribosomal subunits, and translocates mRNA. Ribosomal RNA performs all these roles in the ribosome, the proteins do not. The proteins aid assembly, and folding, and through that catalysis, but are not directly involved in any of the actual functions. And while cells today also employ proteins in translation initiation, experiments have shown translation initiators are not strictly necessary for translation to occur, they merely speed up the process.
So the RNA world hypothesis provides an explanation for these observations by positing that RNA is today performing these roles because by the time fully fledged coded protein biosynthesis had evolved, the process of their biosynthesis had become completely dependent on these RNA-based systems, and hence could no longer be replaced, leaving all subsequent life with this evidence of their ancestry.