Jesus is saying that virtually all the people He was talking to were vipers - cold-blooded snakes only interested in filling their guts.
They were evil and it was impossible for them to speak the Truth. Their hears were evil and so their speech was evil.
In order for these words the Lord spoke not to apply to us, we must repent and change our minds, our thinking, to that of our Lord Jesus Christ.
Our hearts must be transformed by the power of the Word and the Holy Spirit.
Integrity is having a pure heart and being true to it.