Hi Peter, let’s see what the native speaker are going to say on this topic which is rather complex as without any context it will be rather difficult to compare all these verbs. In the meantime, please note that we don’t say “explain to me”. This is a direct translation from many languages such as German and Russian but in English it doesn’t work. Instead, we say “please explain to me”.
Quite a big question. Let me try to give examples - abstain is often used when you decide not to make a decision either for or against a policy. Waive is when you allow someone not to follow a certain regulation. Forego simply means agree to go without something. Relinquish is rather a dramatic word meaning leve a job or post or position. Dispense with suggests / means not to use/need something. Renounce is again a dramatic word suggesting reject as in to stop following a particular policy/belief/faith. These are just a few uses. It would take a long time to give en exhaustive explanation.