"'No one sews a patch of unshrunk cloth on an old garment. If he does, the new piece will pull away from the old, making the tear worse. And no one pours new wine into old wineskins. If he does, the wine will burst the skins, and both the wine and the wineskins will be ruined. No, he pours new wine into new wineskins.'" NIV translation
Jesus uses everyday things that they would have known to explain the difference between the old covenant and His mission. A new piece of cloth will shrink the first few times that it is washed. If the rest of the clothing had been washed many times, it would have already shrank and when the new piece shrank it would rip the old fabric. Likewise, the old wineskin would be more brittle than a new one and would not be able to expand as the new wine fermented and gave off gases. As pressure built up from the gases that were given off, the old wineskin would rupture and all the wine would be lost.
Fasting was a part of the Mosaic Law (see Leviticus 23:27) and the law stands opposed to grace as far as salvation is concerned (They have nothing to do with each other). Applying the Law to a Christian is what is commonly known as "falling from grace" and it has been a favorite weapon of Satan in his war on the church.
"One Sabbath Jesus was going through the grainfields, and as his disciples walked along, they began to pick some heads of grain. The Pharisees said to him, 'Look, why are they doing what is unlawful on the Sabbath?'" NIV translation
Now, we see that the opposition to Jesus and His ministry is becoming more bold. This time the religious leaders asked their question to Jesus personally. What they confronted Jesus with was not the actual Law but was their interpretation of God's Law. It was, in fact, lawful for the people to pick the grain and eat it but it was not lawful to harvest the grain and store the crop.
"He answered, 'Have you never read what David did when he and his companions were hungry and in need? In the days of Abiathar the high priest, he entered the house of God and ate the consecrated bread, which is lawful only for priests to eat. And he also gave some to his companions.'" NIV translation
We notice that Jesus did not argue about the issue of the Sabbath. Instead, He reminded them of King David (who they honored) and the fact that he had broken their laws when he was hungry. His disciples were not eating the consecrated bread but they were still criticizing them.
"Then he said to them, 'The Sabbath was made for man, not man for the Sabbath. So the Son of Man is Lord even of the Sabbath.'" NIV translation
Jesus used this time to educate them about the purpose of the Sabbath. In Genesis 2, God made the Sabbath a holy day of rest for mankind. Although many seem to think that this was created as a day to worship God, it was not intended for that purpose. God knew that the frail human body required a day of rest and so He gave us that day. We are called to worship Him every day and not just on the Sabbath. The religious leaders of Jesus' day as well as many of our day are stuck on legalism and suffer from a lack of understanding of the Word of God.