I've never liked that verse. It is so stark, so harsh, so unnatural. It helps, of course, to know that the language he uses -- hating our families and lives -- is making use of a Semitic idiom that means "to love less." In other words, "I love this and I hate that" meant "I prefer this to that." For example, the Bible tells us that Jacob loved Rachel but hated Leah (Gen. 29:30-31) and that God loved Jacob but hated Esau (Mal. 1:2-3). This does not mean that Jacob actually hated Leah or that God literally hated Esau, but that they preferred one over the other. So Jesus' saying means that families should not be preferred to God. We should love our families, but we should love God even more!
This has some important implications. For instance, the Bible often teaches the importance of obeying or giving deference to those who are in authority over us (Rom. 13:1-7; Eph. 6:1-8, for example), and some evangelical Christians say there are no exceptions to this. According to this perspective, we should always keep our place in the "chain of command" so we will be "under the umbrella" of God's authoritative protection. However, this teaching ignores some scriptural exceptions. For instance, in Ezekiel 20:18-19, God specifically commanded children to disobey their parents! "I said to their children in the desert, 'Do not follow the statutes of your fathers or keep their laws or defile yourselves with their idols. I am the LORD your God; follow my decrees and be careful to keep my laws." In other words, the children were to disregard the teaching and example of their parents because their parents were in rebellion against God. They needed to prefer God, not their families.
Jesus clearly said that faithfulness to him would sometimes result in division within a family (Mt. 10:34-36). He was not diminishing the importance of families, but he was emphasizing that no loyalty can rival our devotion to him.
And because of our commitment to Jesus, we are part of a larger, more-encompassing family. "'Who is my mother, and who are my brothers?' he asked. Then, pointing to his disciples, he said, 'Here are my mother and my brothers. For whoever does the will of my Father in heaven is my brother and sister and mother'" (Mt. 12:48-50).
Our primary identity, then, is as members of God's family (Gal. 6:10; I Pet. 4:17), the "household of faith" (I Tim. 3:15). Jesus says our status is defined by our volitional loyalties, not by the biological or cultural roles over which we have little control. This does not mean that we will not care about our families of origin, but Jesus has radically transformed the meaning of "family" for all those who would seriously follow Him.