"On the contrary, we speak as men approved by God to be entrusted with the gospel. We are not trying to please men but God, who tests our hearts." NIV translation
Paul goes on to explain that they came and proclaimed the gospel because of God's calling on their lives and not the will of men. He goes on to explain that, because God is the One that chose them to do so, it is God that they are out to please and not men. God is the boss and that is who they answered to. Today, as in the past, there are those that think that they have to pretty up the message to make it acceptable to men (after all that is how they earn a living). Paul stands in sharp contrast to that as, even when it was difficult, they did not change the message but boldly proclaimed the Word. Those that God has chosen and approved can trust that God will provide for them and so their "living" does not depend on men but solely on God.
"You know we never used flattery, nor did we put on a mask to cover up greed - God is our witness. We were not looking for praise from men, not from you or anyone else." NIV translation
Paul continues by explaining that they didn't come with tricks or having a hidden agenda and that God knows their hearts in this matter. He goes on to explain that they were not even seeking approval from men but only from God. An example of the impure motives comes from one of my true life experiences.
As part of a "church" we would go out on Monday evenings calling on people that had visited the worship service. On one occasion, I went to do this with an elder of the "church" and he explained that he was doing it because we had to get more people in to help pay the bills. I never participated in the visiting ever again because that is an impure motive.
"As apostles of Christ we could have been a burden to you, but we were gentle among you, like a mother caring for her little children." NIV translation
Paul continues to describe the method that God had them to use in bringing people to Christ. That method was simply speaking the truth in love and it is the same method that God would have us to use today. Many want to simply talk of love and even go so far as to compromise the truth while others simply try to beat down the struggling person and not show them love.