Praise be to Allah, and may peace and blessings be upon the Messenger of Allah.
The basis of this hadith is authentic and is recorded in Sahih Muslim, narrated by Abu Musa al-Ash'ari (may Allah be pleased with him). It mentions that some Muslims will come on the Day of Resurrection with great sins, then Allah will forgive them and place them upon the Jews and Christians.
As for the narration that mentions that Muslims will be divided into three groups, and that the angels will say about those who committed sins, “These are Your poor servants,” and then they will be commanded to place their burdens upon the Jews and Christians and have their good deeds replaced with them, this addition requires verification of its authenticity and should not be relied upon merely because it appears in some books.
As for the meaning of the authentic hadith, it does not mean that a Muslim's actual sin is transferred to a disbeliever so that the disbeliever is punished for the Muslim's sin. This would contradict Allah's statement:
“No bearer of burdens will bear the burden of another.” [Qur'an 6:164]
Scholars have explained the hadith in several ways. One interpretation is that Allah forgives the sins of those Muslims, while the disbelievers are deserving of punishment because of their own disbelief and sins. Thus, they enter Hell because of their own deeds, not because of the sins of Muslims.
Another interpretation is that some disbelievers may have been a cause of Muslims falling into sins, such as someone who establishes and calls people to an evil practice. Such a person bears a burden similar to the burdens of those who follow him, as the Prophet ﷺ said: “Whoever introduces a bad practice in Islam will bear its burden and the burden of those who act upon it after him.”
In conclusion, the authentic hadith does not mean that Allah unjustly punishes Jews and Christians for the sins of Muslims. Rather, Allah forgives the believers, while He punishes the disbelievers for their disbelief and their own deeds through which they deserved punishment.
And Allah knows best.