What is the effect of {} in \eqalign?

\tracingonline1
\showboxdepth\maxdimen

1
$$\eqalign{{}+7=b\cr}\showlists$$


2
$$\eqalign{+7=b\cr}\showlists$$

\bye

a {} just makes an empty \mathord, and the\mathbin from + gets treated like a \mathord unless it is not at the start or end of a list.

In the first case you see

.....\hbox(0.0+0.0)x0.0
.....\glue(\medmuskip) 2.22217 plus 1.11108 minus 2.22217
.....\tenrm +
.....\glue(\medmuskip) 2.22217 plus 1.11108 minus 2.22217
.....\tenrm 7

that is an empty mathord atom, medmuskip either side of the + as it is a mathbin, then 7

In the second case you see

.....\tenrm +
.....\tenrm 7

no leading empty mathord (shown as \hbox) then + and 7 with no spacing as the \mathbin nature of + is suppressed.

Your first example, after the & the {} has no effect as the \eqalign macro adds a {} in the second column already:

\def\eqalign#1{\null\,\vcenter{\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}

so the template for the second column is {}## and already starts with an empty mathord, so there you are comparing {}+7 and {}{}+7 which are the same output.