The gospel of Matthew has a lot of interesting things to say about bread. Its first mention is when the devil tempts Jesus in Matthew 4, commanding the stones to become bread. Jesus refuses to do a bread miracle to satiate his own hunger, despite the fact that he will ultimately feed at least 9,000 men with miraculous bread. Jesus' rebuke of the devil says in essence that physical bread does not sustain the variety of life that God has in store for men, but the words of God only.
Interestingly, In Matthew 7 during the Sermon on the Mount, he says that a loving father does not give his children stones when they ask for bread. So we make the connection: the devil wants Jesus to bypass the complete and total dependence upon the Father and usurp authority, rather than have it granted him. Jesus would have bread, spiritually and physically, but not from his own exercise of power but rather granted by the Father.
By Matthew 9-12, Jesus has extended the metaphor of a grain harvest to his mission. Then the parable of the sower. Then the parable of the leaven.
Finally, Matthew 14, the feeding of the 5,000, with 12 baskets left over. Then Matthew 15, the feeding of the 4,000 with 7 baskets left over. Finally, Matthew 16 where the disciples get worried about bread and Jesus explains that the bread miracles are not really about bread and warns them against the "leaven" of the scribes and Pharisees, which is their teaching.
So these are the clues. Bread is likened to words from God that grant life, like the manna for Israel. Teachings are likened to leaven because they begin small but work themselves through the entire population. The temptation is circumvent the right way to get bread, and through worry and anxiety about having enough to eat to miss out on the true bread.
On a simple level the symbolism of the extra baskets of bread is that you seek the right teaching, the teaching that corresponds to the word of God, regardless of the immediate circumstances and afterwards you find that there is an abundance left over because God is a gracious provider. The significance of twelve? Probably a basket for each disciple (and corresponding to them, a disciple for each tribe of Israel). The significance of the seven? Probably a perfect completion of the provision.
So when the disciples are still concerned about bread, Jesus warns them about the leaven of the Pharisees, the teachings that suppose to be spiritual but are focused on fleshly, immediate, worldly desires at the expense of the truth of God.