What is the significance of the quote from The Crucible, which is listed below?"They [the books] must be [heavy]; they are weighted with authority."
This statement from The Crucible was made by Reverend Hale as he arrived in Salem. These books are the reference books Hale will use to determine whether or not Salem has, in fact, been visited by witches. Reverend Parris makes a rather off-hand comment about how heavy they are as he helps Hale carry them to his house. Reverend Hale sees nothing to joke about and replies with the quote you cite above. This statement has significance in several ways.
First, though these sober Puritans were unlikely to be too frivolous in such matters, Hale's statement reminded them all that witchcraft (and thus the presence of the Devil and his spirits) was no joking matter. Second, Hale was a learned man who had studied these tomes carefully, and the fact that he had been sent for was an indication that the Devil may very well have set up shop in Salem. He was the authority, based on those books, which were another authority. Third, if these books proved the existence of witchery they carried the spiritual weight and authority of law. The books were the authority of the law in terms of these matters; and the keeper (and interpreter) of the books, Hale, was their authority.
In my opinion, the significance of this quote is that it shows the attitudes that people have towards the idea of authority. It shows that people value authority -- they think that what authority figures say is important.
This is in contrast to a kind of time when people value authority less and value evidence or reason more. In our day (I hope) we do not ask "what do the authorities (the conventional wisdom) say about this." Instead (I would hope) we ask "what makes sense here" or "what evidence is there for this."
But Salem was not like this back then. They did not ask for evidence that witchcraft really existed. They did not try to reason things out. Instead, they asked what the authorities thought and they just went with that instead of thinking for themselves.
When people just rely on authority instead of thinking for themselves, they are in danger of having a herd or a mob mentality -- they just all act the same (for example, they persecute witches) without thinking about what they are doing.