"Eat anything sold in the meat market without raising questions of conscience, for, 'The earth is the Lord's and everything in it.'" NIV translation
This passage refers to the fact that the choice meats were offered at the temples of the false gods in Corinth. After they were presented at the temple, they were then taken and sold in the market. Paul reminds them from the Psalms that everything is God's and these false gods are really nothing. Therefore, it is permissible to eat these choice meats that were sold in the market.
"If some unbeliever invites you to a meal and you want to go, eat whatever is put before you without raising questions of conscience. But if anyone says to you, 'This has been offered in sacrifice', then do not eat it, both for the sake of the man who told you and for conscience' sake - the other man's conscience, I mean, not yours. For why should my freedom be judged by another's conscience? If I take part in the meal with thankfulness, why am I denounced because of something I thank God for?" NIV translation
Paul uses the example of a meal and this meat that had been sacrificed in the temple of the false gods to make his point about legality, liberty and love. We see that, as believers in Jesus, we are free to eat whatever but we limit our own liberty because of the love that we have for the unbeliever.
Paul brings up the matter of conscience and that is simply defined as a guiding voice or feeling to tell a person what is right and wrong. This is from God and, for a Christian, this is the Holy spirit speaking to our soul and sanctifying us. Paul asks a simple question that needs to be asked more today than ever before among the "churches" when he asks why his freedom should be judged by another's conscience. The answer is quite simple in that we are all responsible for listening to the Holy Spirit and for what is said to our hearts not what is said to another person's heart. This judgment by so-called "super saints" has caused much damage within the body of Christ and the lives of individuals.
"So whether you eat or drink or whatever you do, do it all for the glory of God. Do not cause anyone to stumble, whether Jews, Greeks or the church of God - even as I try to please everybody in every way. For I am not seeking my own good but the good of many, so that they may be saved." NIV translation
Since we are free in Christ, we should exercise that freedom by letting people see the love of God through us. When Paul speaks of not causing someone to stumble, this has often been misunderstood. Paul is talking about trying to apply the law to people that are saved by grace. He is speaking of the fact that we are saved by grace and nothing that we can do or not do can add to the work that Jesus did on the cross.