- The governor announced [his new tax plan].
- The governor announced [his new tax plan worked].
- The governor announced that [his new tax plan worked].
In the first sentence his new tax plan is the Direct Object of the verb announce. When announce is used with a Direct Object in this way, the Direct Object does not give us any information. We don't know the content of what was said.
In the second example, the bit in brackets is a clause. It represents information and it tells us the content of what was said.
Content clauses like this often have an optional that when they are the complement of a verb:
- He said [he was going to be late].
- He said that [he was going to be late].
In the Original Poster's example, the style advice is to use the word that. This is because it tells the reader that the phrase his new tax plan is the beginning of a content clause and not a complete Direct Object of the verb. After the word that the listener is expecting a whole clause. The person giving the advice is arguing that this makes the writing clearer. It may be true in some cases. I'm not sure if it is true in this particular sentence. Using the word that will definitely not be wrong though.
There is a reason we don't know straight away whether his new tax plan is the Object of announce or the Subject of another verb, unless we use the word that. The reason is that English nouns don't usually have CASE. They are not nominative or accusative, so we do not know if they are Subjects or not. If the word after announce is a pronoun, then we will know immediately:
- The governor announced they ....
In the sentence above we do not need that. Because it is they and not them we know that this is the Subject of a new clause.
Whether you should use that or not depends on the sentence and the context. If people will find it difficult to read easily without that, then use that in your sentence.