Members of the Church of Jesus Christ of Latter-day Saints are familiar with verse 10, and we often quote it in support of the way we offer tithing. In the ancient world, it would be much more like tithing in kind, for they did not have a monetary economy.
Nevertheless, in the context more original to Malachi (and applicable to those in Bountiful), the tithes and offerings were manifestations of faithfulness to the covenants. The ancient tithes supported the priests in the temple. The offerings were the sacrifices to be offered on behalf of the community. It both cases, bringing tithes and offerings were part of faithfully adhering to the human part of the covenant between Jehovah and Israel.
How are the windows of heaven opened with tithes and offerings? Both were destined for the temple, and thus the meaning beyond the specifics is that the temple (or temples, in the modern world) would be able to operate and become the true house of the Lord. In the temple, communication with God increases, and hence God can bless us.
When we live according to our covenants, we can receive the blessings of heaven.