"Ring out, wild bells" starts with a plea for change, addressed to nature itself: "The year is dying in the night; / Ring out, wild bells, and let him die." It pauses on an image of snow, which is often associated with purity (think of the phrase "as pure as driven snow"), but is also harsh on the human body. The poem is informed by Tennyson's extreme grief at the death of his close friend, Arthur Hallam (to whom In Memoriam is dedicated), but it is also an expression of cautious optimism that not all change is bad. We're suffering right now, but purity is not entirely out of our reach.
While the poem begins and ends with thoughts of purity and spirituality, the middle becomes concerned with concrete, tangible human failings. These are impure, largely because they're concerned with material things. Now we're in the world of "party strife," "civic slander," "foul disease," "lust of gold," and "wars of old." Can't you just see a dingy, smelly city street, strewn with trash, as you read those lines?
Among this litany is "false pride in place and blood." In essence, you feel "false pride in place and blood" when you think you're important just because of where you live or who you're related to. Does being born in the United States make a person better than being born in Canada? Does supporting your local football team make you better than fans of other teams? Does having a hereditary title (one that you inherited rather than earned) make you automatically better than people who don't have one? This poem thinks not.
Look at the types of things identified as good in the poem: sweet manners; pure laws; love of truth and right and good; peace; valiance; kindness. What do they have in common? They are things you do. You have to choose to love truth or write better laws or act kindly. It doesn't just happen by accident.
The poem sees pride as false when it's not earned. Being born in a certain place or born into a certain family is arbitrary. There's nothing for you to be proud of, because you didn't do anything.
The reason this false pride is dangerous, and should be rung out, is that it creates needless conflict between people. It can lead to the civic slander and party strife—everything from football hooligans to nationalistic armies to peasant revolts to schoolyard bullying.
In a larger sense, the poem scorns pride in place and blood because it's one of many expressions of our moral failing—we value material things at the expense of spiritual ones. That's why the last line is an expression of hope that Christ will come: if anyone can help us get our priorities straight, it's him.