The gospel can be seen as pearls. Jesus had just said that we shouldn't judge others - the plank in your eye could be a spirit of judgment or condemnation. Instead, think of how God treated us before we accepted him as our lord and saviour. He had a spirit of mercy and grace, which we should extend to those around us. Love the people you're trying to correct.
The problem is not with the gospel message, instead it is that those who haven't accepted Jesus have hard hearts. We are all spiritually dead until God steps in. The approach of forcing the gospel onto someone or manipulating them to agree does not produce a true relationship with Christ and can lead to the opposite outcome (just at the pigs in this passage "trample [the pearls] under their feet, and turn and tear you to pieces.") We can end up driving people away from the gospel by not respecting their right to choose whether or not to follow God.
We must allow people to make this decision on their own. Just as the father in the parable of the prodigal son allowed his youngest son go despite knowing the path the son was on was not best; God honours a person's right to reject him. When you get rejected for trying to share the gospel, it may not be in God's will for them to accept at that time. Sometimes you need to move on and share with others who may be seeking God. This doesn't mean you cannot go back to that other person another time.
Verses 7-12 speak about prayer. You can't force someone to seek God if their heart is hard, but you can pray on their behalf that God may raise them from their spiritual death. Even God does not want to force his good gifts on us, he wants us to choose him. We honour God by trusting that we are asking for good things instead of looking to do it ourselves our manipulating the situation.
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