First of all, let me ask you this: what does He mean by a broken and a contrite heart? To me, this is saying that we, who are prone to set our hearts on things of this world (including people), need to come to Christ with a heart that has been broken by the world, and truly sorrowful for having placed our hope and trust in temporal, and not eternal things.
Luke 15:11-32 : This parable tells the story of what our Father in heaven did, and does for us(I Tim.1:15, Luke 15:20-24). As the father of the prodigal (wasteful)son, ran to his son, so our Father comes to us, in the form of His Son Jesus Christ (Colossians 2:9), and as the father in this parable killed the fatted calf, so our Father killed the Lamb, for our sins. We are born, or were born, sons of God, but, since the Fall(Genesis 3),we are born in sin, and living prodigally (wastefully), like the son in the parable, until the Father, again in the form of Jesus Christ His Son, comes to us (Ezekiel 36:25-27, John 14:6),and we come to our senses and are found worthy, through Christ, to be called sons once more.
Now, the second son, the one who stayed and was faithful (at least outwardly), I think can be likened to those who have their own righteousness(i.e. those who think that by obeying the letter of the law, they will be saved; Luke 18:10-14,18-21). Instead of trusting in our obedience to the law for our salvation, as the Pharisee in Luke 18:10-14, the rich young ruler in Luke 18:18-21,and the elder son in Luke 15:29 did, we should trust in our Fathers mercy to take us back into His kingdom, as did the father in our parable take his wayward son back into his household. Now, you may ask yourself why the elder son was angry, in verse 28. This is explained further, in verses 29 and 30, but I think we may also return to the Old Testament for the reason.
Turn to Genesis 4:1-16. This is a somewhat different situation, but I think it can be applied here as well. I believe that God accepted Abels sacrifice over Cains, not necessarily because of what he did or didnt offer, but in the attitude in which he offered (Psalm 40:6, Proverbs 21:3, Hebrews 10:5-9). Because of the way that Cain and the elder son acted (Genesis 3:5-9, Luke 15:28-30), we can surmise that their condition was present beforehand; that condition being sin. Just like Cain hated Abel because he did not think that Abels sacrifice was any more worthy of being accepted than his, so the elder son was angry and hated his younger, prodigal brother because he, although he had not been obedient all his life like his older brother (Luke 15:29), his sacrifice(Luke 15:18-19,Psalm 51:17) was accepted. So we see that the elder son was angry because his sacrifice of obedience to the letter of the law (verse 29) was not accepted, or so he thought, while his brothers sacrifice(Psalm 51:17, Luke 15:18-19) was, and because, in a way, he was a son of Cain, and his father before him, Satan! The Fathers response, in verse 31, is the same as Jesus response to the wayward disciple in John 12:4-8, and in His response to the disciples of John in Matthew 9:14-15, so let us take a lesson from this, and not be as the elder son, but make our sacrifice as God desires, as the Psalmist says in Psalm 51:17, and as the Apostle Paul said in Romans 12:1, a sacrifice of a broken and a contrite heart and a life of service to Him!
And finally let us thank and praise our Father in Heaven that He has come to us, who were dead, and made us alive in Christ, and sought us when we were lost, and now by His amazing grace, We are found!