The Jefferson "Bible" would be more accurately described as the Jefferson New Testament. Jefferson mashed up/cut and pasted the New Testament to remove any references to the supernatural, or miracles, as well as the divinity of Christ. His title for the book was "The Life and Morals of Jesus of Nazareth," which tells us a lot about his motivations.

He was editing the New Testament to be a compilation of the moral teachings of the man known as Jesus Christ. Jefferson intended to distill the NT down to "the most sublime and benevolent code of morals which has ever been offered to man." Jefferson was a naturalist and a deist, and was not an orthodox Christian in any real way. He considered himself one, based on his own definition that anyone who follows the basic moral teachings of Jesus could be considered a Christian, but that doesn't jive with any formal statement of Christian doctrine. While he believed in a creator god and afterlife, that's about as far as it went. Specifically, his rejection of the Trinity & the divinity of Christ would place him outside the mainstream of both Protestant & Catholic dogma.

Jefferson was also a noted anticlericalist, rejecting the political, social and religious influence of priests/pastors. This plays into his particular apathy for Paul. I'm not going to get into the historicity of Paul, but rather discuss how he and the writings attributed to him were viewed in Christian history, as this was what Jefferson was reacting to.

So let's talk about Paul/Saul of Tarsus (a city in modern day Turkey).

According to mainstream Christian thought, Paul a Pharisee who, following his dramatic conversion to Christianity, became the replacement apostle for the dead Judas Iscariot, despite not having followed Christ in life. Paul was the author of fully half of the New Testament, mostly through his epistles, letters written to various early Christian communities. I feel it's incumbent to add a caveat regarding the term Pharisee here. Within historic Christian thought, the term has come to mean "a legalistic Jewish scholar of religious law." It's a pejorative, often used to imply hypocrisy, and historically was used in a particularly anti-Semitic context (which you can still see today). In reality, Pharisees were nothing more than a specific branch of Jewish rabbinical scholarship that had a variety of social and religious attitudes. Jesus himself could quite reasonably be considered a Pharisee, or at least Pharisee adjacent.

As such Paul played a major role in turning what are essentially rather esoteric but broad moral principles from the teachings of Christ into early Church doctrine. He was the guy who ruled (though he didn't formally hold any position like a catholic pope) on questions like "Should Christians keep kosher? Do non-Jewish Christian converts have to be circumcised?" Paul had an education in philosophy, literature & ethics, including the Stoics, in addition to his rabbinical training under Gamaliel the Elder, a particularly noted rabbi. From a secular point of view, Paul, more than anyone else, including Jesus, founded Christianity as its own fully-fledged, independent religion, rather than a particular Jewish sect that followed a mystic who was crucified by the Romans.

Paul was the guy who took the concepts of Christ and the experiences of the early church & did the worldbuilding and backstory work that provided the foundations for the development of Christianity as we know it today.

To bring us back to Jefferson, given his attitudes regarding priests, deism and denying both the supernatural in general and specifically the divinity of Christ & the resurrection, it's easy to understand why Jefferson held Paul responsible for what he viewed as the errors of mainstream, orthodox Christianity. Accordingly, his motivation for creating his New Testament was to develop a, shall we say, universal set of moral teachings from the writings/saying of Christ, shedding what he saw as grafted in error. The result is a slim volume that contains moral principles and teachings that most people, regardless of religion or lack thereof, could find truth in.

Edit: You can read the whole Jefferson Bible here if you're interested.