First of all a little bit of theory: when TeX finds a character of category 11 or 12 in math mode, it looks at its mathcode; for example, the usual mathcode of + is "202B in hexadecimal form, which means
+ an operation symbol (first digit "2); the character must be taken from math family "0 (second digit); use the character in slot "2B (third and fourth digits).
However, if the mathcode is the special value "8000, TeX looks for a definition of that character as if it were active (catcode 13) and uses that in place of it. Plain TeX and LaTeX exploit this feature for ' in formulas such as $f''(x)$, with a clever definition of active '.
Therefore the problem is to provide such a definition without activating the character. Let's look at the presented code.
\mathchardef\mathplus=\mathcode`\+ % + character in math mode
This line just provides a command that is an alias for the usual +; note that one doesn't need to know the mathcode: that notation access it implicitly.
\mathcode`\+="8000 % set + active in math mode, otherwise inactive
This assigns the special mathcode to +.
\binoppenalty=10000 % do not break elsewhere
This tells TeX that no binary operation symbol is a good point for breaking a line, unless an explicit good break point is manually added: for example a penalty or a discretionary item.
{\catcode`\+=13 \expandafter }\expandafter
\newcommand\noexpand+{\mathplus\discretionary{}{+}{}}
The purpose of this code is to define the active + to expand to \mathplus\discretionary{}{+}{}. One cannot say
{\catcode`\+=13 \gdef+{\mathplus\discretionary{}{+}{}}}
because there would be an active + in the replacement text and this would lead to infinite recursion, because + would be replaced by its definition ad infinitum. Thus the + is activated in a group and, before the group is closed, the \noexpand+ is expanded via the two \expandafter commands. The expansion of \noexpand+ is just an active + that, however, cannot be expanded just now, because its ability to expand has been nullified. Then the group is closed, but the first + (not preceded by \noexpand any more) has already been tokenized and TeX sees
\newcommand+{\mathplus\discretionary{}{+}{}}
where the first + is active and the second one isn't. The purpose of \discretionary is twofold: first of all it inserts a plus sign with the correct mathcode that qualifies it as a binary operation and provides a break point; secondly, if a break is taken, after the break a + is inserted.
A perfectly equivalent definition is
\begingroup\lccode`~=`+
\lowercase{\endgroup\def~}{\mathplus\discretionary{}{+}{}}
because the \lowercase would put back into the main token list
\endgroup\def+
with + of the same category code as ~ (that is, active). The \endgroup undoes the \lccode assignment. The second + has category 12 as usual.
However this definition is not as good as it might seem: try, for example
\parbox{1pt}{\itshape$a+b$}
The \parbox with a very small width is just to force a break after the +; the + at the start of the second line will be from the italic font! Indeed the arguments of \discretionary are processed in horizontal mode, as if \mbox surrounded them (TeXbook, page 287). A more sensible definition would use
\discretionary{}{\the\textfont0+}{}
Since no break can be taken in subscripts or superscripts, this is quite safe. For the - it's another story.
Personal opinion. Some typographical traditions, notably in Russia, use the repetition of operation and relation symbols at breaks. This is a tradition that's worthy of being forgotten as it has no mathematical justification.
