Context is key, so the first place to look is the next part of the same passage:
“Whoever loves father or mother more than me is not worthy of me, and whoever loves son or daughter more than me is not worthy of me; and whoever does not take up his cross and follow after me is not worthy of me. Whoever finds his life will lose it, and whoever loses his life for my sake will find it." — Matthew 10:37-39
Because he is our Creator and Redeemer, Jesus justly demands our highest love. Put another way, we cannot rightly love anything more than him, not even our families or our own lives. This kind of radical love for God is the only thing that can bring peace within ourselves or, ultimately, between people.
However, in the meantime, prioritizing love of God provokes opposition. People, ideologies, governments, and the like don't like being told they're coming in second to anyone, not even God. The total love of God that Jesus calls for can and does cause divisions in families and societies. We should do what we can to avoid discord, but if we subordinate our love of God to anything in order to make peace, we abandon the very foundation on which true peace can be built.