What’s really going on in these three questions is generally lost in English translations, since these translations usually translate two distinct words as ‘love’. So, the conversation in English makes little sense:
Do you love me more than these? (Yes, you know I love you.)
Do you love me? (Yes, you know I love you.)
Do you love me? (You know that I love you.)
However, that’s not at all what’s going on here: in the first two questions, Jesus is asking the question using one word for ‘love’, and Peter answers using a different word. The third time He asks, Jesus uses the word Peter has been using. Peter is not grieved that Jesus has asked the same thing three times – he’s aggrieved because in the third time, Jesus asked what Peter had already answered.
The words used are ἀγαπᾷς (a form of the word ‘agapē’) and φιλῶ (a form of the word ‘philía’). The questions that were really asked, then, were:
Do you agapē-love me more than these? (Yes, you know I philía-love you.)
Do you agapē-love me? (Yes, you know I philía-love you.)
Do you philía-love me? (You know that I philía-love you.)
The question for the exegete, of course, is how we interpret the significance of the difference in these terms as Jesus & Peter used them here, and what significance there is behind Jesus’ change in the question being asked. (I’ll leave that one an open question, although I’ll note that Dave’s appeal to Peter’s denial seems to be spot-on.)
As far as the meaning of “greater than these”, a word search of the ways that similar constructions are used throughout the Bible would tend to lead us to the conclusion that what is meant is “do you love me more than these (love me)?” That leads us directly back to Peter’s assertions, seen literally at Matthew 26:33 (and referenced indirectly in John, through the notion of ‘scattering’ in John 16:32). Peter claims he will not stumble, even though all others might. So, again, the notion that Jesus is referring back to Peter’s assertion of faithfulness and his triple denial – by asking him three times to affirm his love for Him – makes perfect sense here.