It's really not hard: people look to the CDC and the FDA for guidance on the vaccine. At the time, Robert Redfield, CDC Director, said (accurately) that the vaccine would be ready in early 2021. Donald Trump goes out, refutes that, and says that it will be ready in 'late October' (ie. before the election).
In light of that, it's not exactly surprising that people wouldn't find Trump a trustworthy source on the vaccine. My standard on taking the vaccine always came down to buy-in from experts.... and for whatever reason, people just sort of gloss over how a politician (Trump) attempted to interfere with that process for political ends. And how that may have damaged trust.