I thought the order of precedence of operators and quantifiers was arbitrary, but I don't really understand why those three have the same "strength" in relation to other operators (e.g., ¬ will have precedence over ∧, but not over ∀). This leads to the rule being that ¬, ∀ and ∃ will bind to the closest predicate on their right (if I understood correctly). Why is this?
I thought the order of precedence of operators and quantifiers was arbitrary
Certainly the rules of precedence could be set up in different ways, but some ways are more helpful and convenient than others, and there are principled reasons for this.
but I don't really understand why those three have the same "strength" in relation to other operators (e.g., ¬ will have precedence over ∧, but not over ∀).
Assuming you mean "strength" as a synonym for "precedence," the reason these three have the same precedence is that these operators are all prefix operators: as you said, their arguments are the next expressions to the right. ¬ takes the next expression as its single argument, and ∀ and ∃ each take the next two expressions. As long as you use only prefix operators, there is never ambiguity as to which operators govern which arguments: There is only one possible answer to "What's the argument of $\neg$ in $\neg \forall x \forall y P(x, y)$?" As such, precedence rules between these prefix operators would be unnecessary.
On the other hand, when you introduce infix operators such as ∧ (or mix in postfix operators), parsing becomes ambiguous without precedence rules or parentheses. For example, $\neg A \land B$ would be ambiguous because $\neg$ is a prefix operator and $\land$ is an infix operator. To fix this, we could just use parentheses, but in a complex expression this becomes cumbersome. Precedence rules, if agreed upon, allow us to be unambiguous with less notation. By following the rule "¬ will have precedence over ∧", we can write and read "$\neg A \land B$" and agree on what it means.
This leads to the rule being that ¬, ∀ and ∃ will bind to the closest predicate on their right (if I understood correctly). Why is this?
I think the chain of "leading" here is backwards. Because these three are prefix operators (their arguments are the next 1 or 2 expressions to the right), there's no ambiguity between them and thus no need for precedence rules.
Order of precedence is simply a notional convenience. There is no notion of strength here, just notation. All three operators are unary operators with notation "$\circ\ \cdot$", where $\circ$ denotes the operator symbol $\exists, \forall,\neg$ and $\cdot$ the operand. There can never be any ambiguity in which order to apply these operators: the operator to the right must always be applied to the operand first.
Hence, they have the same precedence among eachother if we consider only those three operators. (Note that there can be ambiguity if the unary operators have different position, e.g. $-x^2$, this could mean either $(-x)^2$ or $-(x^2)$ if there was no precendence between $^2$ and $-$.)