What does it mean to "rightly divide the Word", and how do we do it?

It is crucial for every believer, when reading the Bible, to rightly divide the Word, and to clearly separate what belongs to the old covenant of law and what belongs to the new covenant of grace. When people quote Old Testament passages without appropriating the cross of Jesus in their interpretations, they make it seem as though the cross of Jesus Christ made no difference at all, leading to much misunderstanding and misinterpretation of the Bible.
What Jesus did on the cross created the new covenant, and made the first covenant—the covenant of the law—obsolete (Hebrews 8:13). In this new covenant, Jesus has already fulfilled for us the righteous requirements of the law (Romans 8:3–4), so that we are no longer under law but under grace (Romans 6:14).
Hence, we need to learn to rightly divide the covenants—by studying and reading all scripture in the light of the cross. This was the apostle Paul's advice to his young apprentice, Timothy: "Be diligent to present yourself approved to God, a worker who does not need to be ashamed, rightly dividing the word of truth." (2 Timothy 2:15)
Whether interpreting the Old Testament, or the words which Jesus spoke in the four gospels (Matthew, Mark, Luke and John), let Jesus and His finished work at the cross be the key to unlocking all the precious gems hidden in God’s Word. This means that we have to read everything in the context of what He came to do and what He accomplished at the cross for us. For example, some things that Jesus said in the four gospels were spoken before the cross—before He had died for our sins—and some were said after the cross—when He had already won our complete forgiveness and rightfully given us His righteousness. It is the latter that applies to us (believers under the new covenant) today.
In rightly dividing the Word, we also have to take note of who Jesus was addressing when He spoke. With the Pharisees, who boasted in their perfect law-keeping, Jesus spoke of the law at its most pristine standard, such that it was impossible for any man to keep. He did it so that man would come to the end of depending on himself and begin to see that he desperately needs a Savior (Galatians 3:24). But to the sinners, the prostitutes and tax collectors, He was never harsh, and was full of compassion for them.
Once you understand how powerful this principle of letting Jesus be the key to understanding and applying God's Word to your life is, you will no longer be troubled by obscure passages in the Bible. This is because the Lord has given so many clear, explicit portions of Scripture that declare His favor and blessings over your life in the new covenant.