"Looking diligently lest any man fail of the grace of God; lest any root of bitterness springing up trouble you, and thereby many be defiled;" (Hebrews 12: 15)
Why do people get bitter?
They fall from grace.
Now, how do we fall from grace? By sinning? Not at all.
By trying to keep the law, by trying to justify ourselves rather than resting in His justification for us:
"I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me." (Galatians 2: 20)
"Christ is become of no effect unto you, whosoever of you are justified by the law; ye are fallen from grace." (Galatians 5: 4)
The law brings out bitterness, as it awakens other sins in our lives.
Why does the law make us bitter, anyway?
"19Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. 20Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin." (Romans 3: 19-20)
People may try to keep the law, yet they find themselves unable to. Those who insist on trying, will themselves doing so in vain, accomplishing nothing. How frustrating!
"56The sting of death is sin; and the strength of sin is the law" (1 Corinthians 15: 56)
The law brings out in the sin in our lives, rather than offering us a means for us to make ourselves righteous. All of that trying, and accomplishing nothing, would make anyone bitter.
Now, how do we get rid of bitterness in our lives? Receive more of God's grace!
"31Let all bitterness, and wrath, and anger, and clamour, and evil speaking, be put away from you, with all malice: 32And be ye kind one to another, tenderhearted, forgiving one another, even as God for Christ's sake hath forgiven you." (Ephesians 4: 31-32)
The whole release from bitterness hangs on our understanding of "as God for Christ's sake has forgiven you."
The more that we understand how gracious He is to us, then we are freed from bitterness, as Christ grants us His "so much better".