To “eat with offense” in Romans 14 means to eat in such a way as to cause others offense and make them stumble in their walk with God. Let me explain.
To understand the verse (vs 20), we need to place it in the broader context of the passage. Romans 14 presents a picture of living together in a diverse Christian community. There are differences in opinion over some issues, which Paul calls ‘disputable matters’ v1. I take it this means that in some areas of day to day living, the christian choices we make will be a matter for the conscience of the individual believer who may have different interpretations of the Bible.
In such matters we are all accountable to God, who is our ultimate master and we ought not stand in judgment over other believers for doing things we can see as ‘disputable matters’. The gospel of Christ sets us free to act in accordance with the will of God (Ro 12v1-3).
Love for our brother or sister in Christ, coupled with knowing God is our judge, is to be what guides us in deciding what is the best way to act in such situations. We may be free to what we please but if what we do causes others (who think differently) to be offended or (even worse) tempt them to do what they think is wrong, then my love for them should make me stop what I am doing. I don’t want to cause them offense.
Food, particularly that which is sacrificed to idols, is one of those “disputable” issues in the early church and is still an issue for many cultures today. If someone believes or has believed in the power of idols or spirits it may be unhelpful for them to get involved in eating food sacrificed to idols. Ultimately you do not want anyone to go against their conscience before God because of a ‘disputable matter’.
Modern examples of this principle in action may be deciding not to drink alcohol because you are with an ex-alcoholic or a Christian who believes in prohibition. It is not that there is anything wrong with alcohol (in moderation - drunkeness is prohibited by scripture in Eph 5:18), the issue is that your brother or sister in Christ may be led into sin by what you do. In that case it is better not to exercise your freedom to drink. If you feel you need a drink in this context you are better off talking to them before rather than assuming that it doesn’t matter. A good corrective is also found in Ro 14:16, where we ought not let the good gifts of God be spoken of as evil.
In matters of right and wrong, however (not disputable things), we are required to correct our brothers and sisters with love and gentleness, using the Bible and other Christians that God has placed around us to help. But where the matter is disutable, we must recognise that there will be differences of opinion and respect them and those who hold them.
Ultimately we are all accountable to God and not to each other. Love will always seek the best for others and not need to force others to agree with it all the time.
I hope this helps you understand this issue.