This proverb is taken from Chapter 3 and is spoken by Okonkwo to explain why he has sought an audience with Nwakibie as a young man without a penny to his name and needing to make something of himself. Note what he says:
I have brought you this little kola. As our people say, a man who pays respect to the great paves the way for his own greatness. I have come to pay you my respects and also to ask a favour. But let us drink the wine first.
Okonkwo uses the proverb therefore to flatter Nwakibie, and also to anticipate the request he will make. The proverb reflects the reality of the world and can be summarised as meaning somebody who respects his betters will allow himself to be advanced in life through that respect. It is the use of this proverb, in part, that allows Okonkwo to persuade Nwakibie to give him the yam seeds he requests so that he can establish himself in his village and society. Okonkwo's story proves the truth of this proverb, as he has respected his betters in order to pave the way for his own advancement in society.
a man who pays respect to the great paves the way for his own greatness
has a twofold meaning in Things Fall Apart. The first meaning is the literal one, where we witness Okonkwo using the proverb to request to meet Nwakibie. He needs a favor from Nwakibie and he is speaking as a weaker, poorer, less important man. Therefore, prior to him having to request something from Nwakibie, he will demonstrate humility and recognize that Nwakibie is, at his point, the stronger and more capable of the two given the circumstances. Hence, here the proverb serves as an act of humbleness.
The second meaning is more universal. Life is like a see-saw. People may be at the top of their game at some point, and for a long time. But there will be a time when they will fall and touch rock bottom. It is inevitable: if it is bound to happen, it will.
If we do not recognize our weaknesses and limitations we will live a life rife with over-confidence, oblivious of reality. Whenever the fall comes, we will fall apart not knowing who we are in the first place. Those who have either come from the ground up, or at least understand what it means to come from the ground up have already learned the steps to get to the top. They are bound to be great because they have conceded that they are not reigning life supreme: they are humans. These are the people who will have an equal chance to succeed in life and, once they reach the top, they will know how not to fall hard again. Their way to greatness has been paved with effort, humility, and knowledge.