What do jurists mean by a man being Muslim by claim and affiliation?

Islam is established through explicit declaration of the two testimonies (Shahadatayn), or by dependency, such as a young child of a disbeliever embracing Islam through his father's conversion, or by implication through performing acts like the pillars of Islam. The fundamental way to establish Islam is by uttering the two testimonies and adhering to their meaning, as the Prophet (peace be upon him) said: "I have been commanded to fight against people until they testify that there is no god but Allah and that Muhammad is the Messenger of Allah, and they establish the prayer and pay the Zakat. If they do that, their blood and their wealth will be protected from me, except for the rights of Islam, and their reckoning will be with Allah."

A person who says "I am a Muslim" is considered Muslim, even if the utterance of the two testimonies was not heard from him. Scholars have agreed that if someone who claims to be Muslim commits an act that nullifies Islam, their claim of affiliation does not benefit them. Sheikh Abdul Aziz bin Baz said: "Whoever commits an act that nullifies Islam, he is judged by that nullifying act, even if he says: 'I bear witness that there is no god but Allah and that Muhammad is the Messenger of Allah,' and even if he prays and fasts. This testimony is beneficial if its rights are fulfilled. But if its rights are neglected, it does not benefit its utterer."