One of the central tenants of the catholic priesthood is abstinence.

That's just not true. In the Latin rite celibacy is generally required (but can be dispensed with on a case-by-case basis, usually in the case of formerly Anglican or Protestant clergy that have converted to Catholicism) but Eastern Catholic Churches are fine with married priests (as long as the marriage has taken place before ordination).

Also it's quite a leap from "doesn't believe he will end up in hell for extramarital intercourse" to "has to be a secret atheist".

First, as others have pointed out, there are exceptions to clerical celibacy, both historically and today. My body of knowledge is mostly restricted to the early modern period; however, this is less of a disadvantage than it might seem at first.

Enforcing clerical celibacy was a constant struggle for the Church, whose (despite what others will tell you) formal position always tended towards clerical celibacy; the doctrine has its roots in Paul's epistles, which state that celibacy was a preferable state to marriage. Celibacy among prelates was definitely the norm, and there was some wrangling in the Late Empire and Early Middle Ages over how celibate lower orders of clergy needed to be. At this point, East and West appear roughly similar.

However, the Roman Catholic Church took a rather dramatic turn in the eleventh century, when clerical celibacy became an absolute in canon law during the height of the Gregorian "Reformation". The next six centuries would see this law enacted to varying degrees over Catholic Europe, aided in part by the impetus of the Protestant Reformation.

Moving more to your questions, not answering particularly in order: since celibacy was now canon law, how could the laypeople accept clergy that flagrantly broke the law, especially bishops and even Popes? The reality here is that Church law very much outpaced the sensibilities and beliefs of the average Catholic. The average Catholic during the Middle Ages and the onset of the Reformation likely attended Mass only occasionally, if that; the Fourth Lateran Council (1215) felt the need to command that all Catholics go to Confession at least once a year.

The short answer to why they put up with it, really, is "the average Catholic didn't know, or particularly care to know." When they did take issue with what they perceived to be corruption, the focus was largely upon the significant economic costs of supporting the Church, rather than the clergy's sexual morality (although Protestants and dissidents would certainly seize on the idea of the lecherous cleric in propaganda). Ultimately, Protestant Reformers would simply do away with clerical celibacy as an unnecessary and unnatural trapping of Rome.

As for justification: many priests may have disagreed with the Church's canons about celibacy, and/or found avoiding such sin unfeasible and undesirable. Often the conceit was that priest's concubines were "cleaning women" or house-servants, even when it was very obvious the two were effectively married.

But there were literally thousands of priests documented as living a carousing lifestyle. Were the all just secret atheists? Or did they think they were going to hell or what?

No, they were simply people. Like with all groups of people, many priests failed to meet society's standards for their behavior. It's worth pointing out here that many priests and clerics were second or third sons with no other real prospects: a career in the Church was effectively mandated for many of these men, and many (if not most) did not feel called to the task. This isn't to suggest that many didn't also perform their roles earnestly or believe sincerely, but they were simply unwilling to make the sacrifice asked of them by the Church.

Catholics, by the way, don't go to "Hell"; most are destined to Purgatory, where they are expiated until they are pure enough to go to Heaven. This lack of a "fail-state" for Catholics, who are not obliged to constantly prove their faith as are most Protestants, means that laxity among both clerics and parishioners is, literally, more forgivable.

Last note: Catholicism also isn't about following "the word-for-word Bible". As others have pointed out, clerical celibacy is not a Biblical doctine—neither, strictly speaking, is the clergy itself. In Catholicism, established interpretation and canon law are the primary guides for Catholics, not an individual reading of the Bible as in many Protestant sects.