When you want to discuss something that somebody has said, it is called "reported speech" or "indirect speech," as opposed to quoting somebody directly. 

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Imagine you are at work, for instance, and a supplier named Daniel tells you "Most of our new accounts are getting a 30% increase, but I can cut you some slack." This means that Daniel will have to charge your company more money, but he can "cut some slack," meaning he can make the increase not so large for your company. When your boss asks you what Daniel said, you would use "reported speech" to tell him: 

Daniel said most of their new accounts are getting a thirty percent increase, but he said he could cut us some slack.

Captions 29-30, Business English - Difficulties with Coworkers and Contracts

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Note the difference in how the speaker changes from first person "our" and "I" to third person "their" and "he":

Direct speech: Daniel said, "Most of our new accounts are getting a 30% increase, but I can cut you some slack."

Indirect speech: Daniel said most of their new accounts are getting a 30% increase, but he said he could cut us some slack. 

Note that indirect speech eliminates the need for quotation marks. Another primary feature of indirect speech is using a phrase such as "he said," "she said," etc. followed by a description of what the person said. Here is a sample of other verbs you can use to report what somebody said: 

—to tell: 

They told me I was going to lose the fight.

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—to state:

A spokesman for the Ministry of Plenty stated last night that it will be necessary to reduce the chocolate ration to twenty grams in April.

Captions 12-13, George Orwell's Nineteen Eighty-Four - BBC TV Movie

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—to mention: 

You mentioned you're single.

Caption 10, Conan - Alice Eve Explains Differences Between American & UK Dating

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–to confess: 

You've confessed to assassination, to distribution of seditious pamphlets, to religion, to embezzlement of Party funds, sale of military secrets, sabotage, murder. 

Captions 28-31, George Orwell's Nineteen Eighty-Four - BBC TV Movie

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—to claim

They claimed to be weavers of a rare and especially beautiful and precious cloth.

Caption 24, Fairy Tales - The Emperor's New Clothes

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Further Learning
Try taking the sample sentences above and reconstructing the direct speech. For example, "He told me he'd come to demolish the house." Change that into "He said, 'I've come to demolish the house.'" Now try it with the other examples!

If you are learning English in a small group, have one person state something as direct speech and another person then report what that person said. If John says "I speak the best English in the class," then Jenny can say "John claims to speak the best English in the class." 

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You can also go to Yabla English and find other examples of indirect speech based on the verbs listed above.