why is a nullary operation a special element, usually 0 or 1?

An operation $\lambda$ with $n$ arguments, that is a $n$-ary operation, on the set $S$ is a function $S^n\to S$. Now we can view an element $x$ of $S^n$ either as an $n$-tuple or as a function $x:\{1,2,\ldots,n\}\to S$. Specializing to the case $n=0$, an element of $S^0$ can be seen as a function $x:\emptyset\to S$. There is exactly one such function that trivialy maps all elements of the empty set to elements of $S$. So $S^0$ has only one element. In the usual set-theoretic way of identifying a function with its graph, we have $S^0=\{\emptyset\}=\{0\}$.

So a nullary operation $\lambda:S^0\to S$ is determined by the value $\lambda(\emptyset)$, a single element of $S$. For convenience, we sometimes treat the nullary operation as if it would be this element $\lambda(\emptyset)$, and in this sense a nullary operation is a constant.

Yes, a nullary operation is an operation that accepts zero arguments. Such an operation in the context of an algebraic structure on a set $A$ must thus produce, for every choice of no arguments, an element in $A$. Since there is precisely one choice of no arguments a nullary operation boils down to a choice of an element from $A$. In many algebraic systems common 'names' for such elements are $0$, and $1$ (or $e$) but it can be anything at all.

It is helpful to consider constants as nullary operations particularly in the context of universal algebra simply since it unifies the things: An algebraic system in universal algebra is a set together with a bunch of operations (of various arities) satisfying axioms.

A nullary operation is an operation that takes no arguments and then produces an element of the set where the operation takes its values (usually this is the algebraic structure under consideration). Since the value of mathematical operation can only depend on the arguments of that operation, and there aren't any of them in this case, this value has to be the same element each time the operation is applied. Therefore specifying a nullary operation amounts to the same thing as specifying the value obtained by applying it.

This value does not have to be a special element. But there is hardly a point of introducing an operation in the language of a type of algebraic structure, be it a nullary operation or one of nonzero arity, if there aren't any axioms relating to the operation. So in practice there will invariably be, for any operation in the language of the structure, at least one axiom of the structure in which the operation is mentioned; in the case of a nullary operation this axiom endows the element designated by that operation with a special property. It is this property that makes this element special. Calling the element $0$ or $1$ just is an aid to remembering what kind of special property this element is endowed with.

It is often customary to give operations of various arity special notations.

For example, the theory of Abelian groups has operations $z, n, p$ of arity 0, 1, and 2 respectively. But rather than write $p(x,y)$, we usually use the notation $x + y$ instead. Rather than write $n(x)$, we use the notation $-x$. And rather than write $z()$, we use the notation $0$.