This is a badly-worded phrase that means nothing in the context of the paragraph in which it occurs. There is no way the reader could be expected to understand it from this awful book on its own. Normally I would not think it the role of this site to remedy deficiencies in text books. However, as this involves some interesting questions I will try to read the author’s mind.
The author probably means something like:
“splicing is a relic of the RNA world”
…although a more measured statement would be:
“the existence of self-splicing supports the idea that the current DNA/protein world was preceded by an RNA world”
The author’s use of the word “predominance” is just incorrect English. (Sloppiness or deficient vocabulary.)
An RNA world?
So what is all this about an ‘RNA world’? The author refers to this on p. 104. He states as a categorical fact what it is important to realize is only a theory:
RNA was the first genetic material…RNA used to act as a genetic material as well as a catalyst…
Why is splicing relevant to this? Some types of splicing are catalysed by the RNA itself — self-splicing — an example of an RNA enzyme (ribozyme). From the existence of these and other ribozymes it is argued that originally all enzymes may have been ribozymes, and protein enzymes only developed later, taking over most, but not all enzyme functions.
It is important to realize that RNA preceding DNA, and the first enzymes being RNA rather than protein, are two independent ideas. Although both these ideas are widely held, and I personally tend to favour them, certain scientists have argued strongly against them, as discussed by Harold S Bernhardt.