In most cases, the meaning of a word can be identified based on its context. In the case of the word "omit," this is used to explain the action of excluding or leaving out something. This can be inferred because the sentence affirms "Some speakers decide to omit details that do not support their claims" and we know details that are not connected to the claim should be avoided or eliminated. Therefore, the word "omit" acts as a synonym of eliminated or exclude and thus, the option that best defined it is "leave out".
