Viral mutation rates and replication
Since no one else has done so, just gonna expand on what occurred here in the comments (mostly from @Alex Reynolds):

tl;dr: natural selection exists.

Longer version:


Mutations/replication errors/whatever will happen in the creation of new virus particles within a host cell. There is some distribution of number of mutational events that will occur as a function of the number of new RNA molecules created per cell and the replication error rate. Some subset of these RNA molecules will be packaged into new particles, yielding a new generation of viruses that get a chance to infect. Yes, this is potentially a large number of new mutations per infection event.
Some large proportion of these new particles will contain non-functional RNAs, due to inactivating mutations of one kind or another. Most of the new mutations will be worse at being a virus than the originally infecting virus (their immediate parent). So therefore, they will be outcompeted by non-mutated forms.
Among all those viruses which continue successfully infecting throughout the course of a year, on average we observe 24 substitution mutations in the genome over the course of that year. Every transmission chain included in the group of viruses at the end of the year must have remained viable for infection of hosts. In other words, it has survived natural selection.


In population genetics, this phenomenon is known as "mutation-selection balance"- e.g. new mutations arise in the population at a rate that is limited by the action of natural selection (at equilibrium).

Genotype-phenotype maps

What of course I am leaving unstated is exactly why "...new mutations will be worse at being a virus than the originally infecting virus..."

This is very hard to predict in many cases. In some cases it is very obvious why a mutant is bad at being a virus, if e.g. an essential gene is deleted from the RNA genome. But for many single mutations we don't really understand what their effect will be on virus function, we just observe that they drop out of the population due to negative selection. 

If you are interested in this question, I recommend looking into genotype-phenotype maps. I know Jesse Bloom has done some work on influenza in this field, here is one example of such a paper. It is generally extremely laborious to do this kind of work, so relatively little is known about the functional effect of any specific mutation in any specific genome.

Hope it is helpful to have all that written out.