Basically, imagine you have three candidates: A, B, and C, and there are 50 voters, and the votes come in like this:

  100 - A > C > B
  100 - B > C > A
  40 - C > A > B
  10 - C > B > A

After round 1 of IRV, candidates A and B each have 100 votes, C has 50. C gets knocked out, 40 votes move to A, 10 votes move to B. A now has 140 votes, and B has 110, and A is declared the winner.

The problem is that the B > C > A don't get their second choice considered. Why does this matter? Look at it when you put the votes into pairs.

  A vs B: 140 people think A > B.  110 think B > A.
  A vs C: 100 people think A > C.  150 think C > A.
  B vs C: 110 people think B > C.  140 think C > B.

C is actually the more popular candidate, but C LOSES using IRV!