Usually when explaining IRV, advocates spend a great deal of time convincing you that the candidate who receives the most (plurality) votes is not necessarily the best. Then, usually without any explanation, they assume that the candidate with the least number is the worst.
The trick (the same kind of misdirection used by magicians): They assume that as a former believer in plurality (and that's almost everybody) you started with this assumption.
There's nothing wrong with eliminating candidates. But eliminating them based on the assumption that the lowest plurality candidate is the worst doesn't make sense, unless you actually believe that first-preference support is the best indicator of popularity. But if you believe that, you can't agree with IRV!
If you believe in IRV, then presumably you'd want to select the losers to eliminate, also using IRV. But that simply will not work, because the IRV winner is entirely capable of being the same person as the IRV loser! In other words, IRV is not self-consistent.
In contrast, Range Voting is logically self-consistent in this respect: if you eliminate the worst (in terms of their range-vote scores) candidates, you end up with the range voting winner.
This argument is extracted almost verbatim from a statement by Blake Cretney.