Ultimately, the speechwriter needs to understand not just what the speaker thinks about the issues, but how the speaker thinks about the issues. In other words, the speechwriter needs to be clear on why the speaker thinks the way he does (i.e. personal experience, values, evidence from history, polls, studies, or research). Speechwriters often find themselves referencing back to this type of supporting evidence as they write.
For longer or more important remarks, it’s a good idea to solicit guidance from the speaker on the front end. Ask about key themes, phrases, stories, or jokes the speaker thinks are relevant to the topic at hand—this will end up being much easier than completing a draft and being asked to work in a new theme later!