In the immediate context, after saying that he will give the keys, he gives the keys. He is the Christ, and he showed them where in the scriptures it spoke of his death and resurrection.
This is obfuscated by historical interpretations.
What is the kingdom of heaven? If it is the 'club' called the church, with a priesthhood which has conquered the laity (Nicolaitans), then naturally, it would appear that the keys are those things which give one authority.
The kingdom is like leaven, which some people say leaven is sin. But the Hebrews got rid of leaven before Passover, which is a well accepted symbol of the cross. One does not rid oneself of sin before coming to the cross. Leaven is teaching. It starts off small and spreads. We are to beware of the teaching of the Pharisees. And there is a theme about getting rid of the old to make room for the new in the parable of patching old garments with pieces of a new garment, and putting new wine in old wineskins.
Mt 16:19 And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.
Mt 16:20 Then charged he his disciples that they should tell no man that he was Jesus the Christ.
Mt 16:21 ¶ From that time forth began Jesus to shew unto his disciples, how that he must go unto Jerusalem, and suffer many things of the elders and chief priests and scribes, and be killed, and be raised again the third day.
The keys to the kingdom/teaching are the same as the outline notes to the sermon on the road to Emmaus. They are the pictures of Christ hidden in the Old Testament, which Paul calls the "mystery hidden from the beginning". Look for the symbols of the cross in the ancient prophecies, then begin teaching around them as they speak of Christ.
Here are some helpful steps:
1. Keys to the kingdom (look for hints of the cross)
2. "Rightly divide the word" Look for the things before and after the cross within the pericope
3. Translate from Greek to Hebrew if necessary. (if working in the NT)
4. Solve riddles. Reword synonyms.
5. Apply the Rules
6. Fill in gaps with deeper digging using correlation (remez and derash) and notarikon
7. Reword the scripture with Christ being central
Here are some keys:
1. Being cast into water (subject to the word of God fully)
a. The ark
b. The axe of Elisha
d. Peter as he dives off the boat
2. Crossing water
a. Red sea
b. Jordan river
c. Elijah-Elisha crossing the stream
d. Jesus asleep while crossing in a boat
e. Jesus walking on water
3. Death of the Christ type in a pericope
a. Cain and Abel/Seth
b. The death of Moses
c. The death of Samson
4. Threat of death
a. Jacob wrestling the angel
b. Joseph in the pit
c. The dreams of the baker and cup bearer
5. Being in prison
b. Ten brothers of Joseph
6. Second sons in fact or riddle
a. Adam in and out of the garden
c. Abraham (check ages)
f. Judah (4th but Simeon and Levi don't count)
i. David (second king)
7. Sacrifices (all of them)
b. water parted
c. Stone split
d. Veil rent
e. Baby threatened to be chopped in half
e. Samson slept 3 times
10. Water in wells
c. At Sychar