According to a paper by the IRS you should declare illegal income from dealing drugs or stolen property. What's the motivation behind this? And are there examples of people who reported such activities and how they got away with it?
The reasoning behind this is simple, everyone is expected to pay tax on all earned income. Actions that may have been illegal such as drug dealing do not exempt someone from paying their fair share. The US Supreme Court ruled that this was NOT a violation of the 5th amendment because they are only asking about income, not the source of that income. You can read about the finer points of this decision in GARNER v. UNITED STATES
As for examples of people reporting illegal income and then getting caught, I can't find any. The people who do illegal things and report their income to the IRS are career criminals. They have a tendency to stay quiet about these types of things and I don't expect many to speak up lest it draw the attention of the authorities. If you know of any, please let me know.
They don't believe you will declare your illegal income. It's a way to stack more potential crimes and charges on those who are criminals. Sometimes you'll get the weird result of the criminal being found not guilty on major criminal charges, but then getting found guilty of not paying taxes on criminal income, despite being cleared on the other charges.
Al Capone, one of the most notorious and violent criminal mob bosses of the Prohibition Era was never convicted of the many murders or other violent crimes he ordered or participated in. He went to jail for income tax evasion.
Capone’s successful conviction set a precedent for future law enforcement officials. Tax evasion would become a popular way to convict participants in organized crime if more substantial evidence was not available.
It's also a way for law enforcement to "put the screws" to those they deem criminals, since they are usually able to confiscate assets deemed to be from illegal activities, again, even if those illegal activity charges, themselves, don't bear fruit in the form of convictions. Getting a conviction on a charge like this, or even getting an indictment makes it easier for them to fight attempts to reclaim assets. The courts have been pretty open to the idea of "we didn't convict them, but we don't think they can legitimately account for this money, so we're taking it away."