This line actually occurs at the beginning of Act 4, Scene 3. It is spoken by Claudius to two or three attendant lords. He says that it is very dangerous for Hamlet to go free (after killing Polonius) but he cannot imprison him because

He's loved of the distracted multitude,

Who like not in their judgment, but their eyes . . .

In other words, the ignorant, emotional masses form their opinions of prominent people on the basis of their looks and manners rather than by evaluating their real characters. Hamlet is young, good looking, graceful, and charming. Ophelia describes him as

The courtier's, soldier's, scholar's, eye - tongue - sword,

Th' expectancy and rose of the fair state,

The glass of fashion and the mold of form,

Th' observed of all observers . . .        (3.1)

Claudius is expressing a truth which is perhaps even more obvious today than it was in Shakespeare's time. There are many men holding public office in the United States, and many more seeking public office, whose main qualification seems to be that they are handsome and have very attractive smiles. They wear their clothese well and pay a lot of money for their haircuts. With the advent of television, good looks became a great advantage for an ambitious politician.

The masses tend to judge by appearance and emotion. They will be supporting Laertes in his insurrection in Act 4, Scene 5 for the same reasons--he is handsome, dashing, aristocratic.