Use the word "claims" to describe what someone said, when you don't believe it or you're not ready to accept that it's true:
The suspect claims that he was at home at the time of the murder.
She claims that I agreed to sell it to her for $300, but that's bullshit. I never agreed to that.
In this example, the speaker used "claims" instead of "claimed". The son said it in the past, so why doesn't she say "claimed"? It's because the son still hasn't changed what he said. He said that he didn't do it, and if you ask him again he'll still say that he didn't do it. You can use "says" in the same way:
He says that he didn't do it.
He swears that he didn't do it.
("Swears" means "says very forcefully" or "promises".)
As with lots of other "that" clauses, people sometimes leave "that" out of the sentence in casual spoken English, like in the example at top.