Squealer, like the propaganda arm of the Russians he symbolically represents, is a convincing speaker who tells the animals exactly what they want to hear--even if it's not true. He projects Napoleon as a great leader whose every move is based on the betterment of the farm and the animals' lives. He constantly reminds the animals that things can only be worse if Mr. Jones returns, and he often uses words that the animals do not understand but willingly accept.
... Squealer spoke so persuasively... that they accepted his answers without further explanation. (Chapter 5)
Indeed, the animals
... always found themselves in agreement with the one who was speaking at the moment..., (Chapter 5)
so once Snowball is eliminated, it simply becomes a matter of Squealer emphasizing Napoleon's wishes. As for Napoleon, he "had a reputation for getting his own way," and his "fierce" nature speaks louder than his words. Squealer promises the animals that "Napoleon is always right," and he knows that the simple-minded animals will believe him.