People angry with the high prices were protesting.
Can we use adjective after noun without verb to be like the example above?
Can you tell me what grammar rule is it?
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As J.R. says, that construction is grammatical, and indeed required.
A bare adjective, or one modified by one or more preceding adverbs, goes in front of the noun. (I'm adding a determiner, many, to your sentence, to make the structure a little less ambiguous.)
Many angry people were protesting.
Many passionately and vociferously angry people were protesting.
But an adjective which has a complement cannot be placed in front of the noun; it must be treated as a reduced relative clause† and placed after the noun.
∗ Many angry with the high prices people were protesting.
Many people angry with the high prices were protesting.
If the adjective is modified it will carry its modifiers with it.
Many people passionately and vociferously angry with the high prices were protesting.
The same thing is true of any ‘heavy’ modifier with an embedded complement, such as a participle phrase or an adjectival preposition phrase:
∗ Many suffering from hunger people were protesting.
Many people suffering from hunger were protesting.
∗ Many from the surrounding villages people were protesting.
Many people from the surrounding villages were protesting.
As Laure says, the clause may be bracketed with commas. This changes the meaning, however: the clause is now ‘non-restrictive’ rather ‘restrictive’: being angry with high prices no longer defines the people who were protesting, it is an additional observation about them. A non-restrictive clause can be placed in other places:
Passionately and vociferously angry with the high prices, many people were protesting.
Many people were protesting, passionately and vociferously angry with the high prices.
∗ before an utterance marks it as ungrammatical.
† A reduced relative clause is one from which the relative pronoun and any immediately following copula have been deleted as unnecessary:
who were angry with the high prices.