I used to see Jesus' statement as a demand of a high standard that anyone has to fulfill that wants to be His disciple. But one day reading it again, I realized that it is much more than that. As long as we hold fast to anything else, sooner or later this always will compete with the importance that Christ has in our thoughts and gain the position, that only belongs to Him.
Reading it today once again, I saw another aspect of this statement: It is one of the tests that God gives us to determine whether we are saved or not. Anyone "who does not renounce all that he has" can not be a disciple of Christ, can not be saved. How many of those in the western hemisphere that call themselves Christians does this throw out!
To look at it in the picture language of Jesus' parable of the pearl: If you truly found that one exceedingly precious pearl, you gladly give up anything else you have to get it. If you are still trying to hold on to everything else and add salvation to it, you cannot have found the real pearl nor know its value.