How do I make a rectangular plot with annotations?

Use width and height to set the dimensions of the axis, adjust to your liking. For the labels, you can first add a coordinate at an appropriate pos at each \addplot, e.g. \addplot {..} coordinate[pos=0.6] (foo);, then add clip mode=individual to the axis options, and finally use \node [pin={30:$C=x$}] at (foo) {}; to make the pin with label.

In the code below I used a loop, as the code becomes a lot shorter, but you can of course write out everything if you find it clearer.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}

\tikzset{
point/.style={circle,draw=black,inner sep=0pt,minimum size=3pt}
}
\pgfplotsset{
    soldot/.style={color=blue,only marks,mark=*}
    }
\begin{document}
\begin{tikzpicture}[every pin edge/.style={black,thin}, pin distance=7mm]
    \begin{axis}[
            % set size of axis
            width=7cm,height=10cm,
            % disable clipping of \node, \draw, etc., while keeping clipping of \addplot
            clip mode=individual,
            xmax=1.5,
            xmin=-1.5,
            axis equal,
            xtick={0},
            ytick={-2,-1,1,2},
            axis lines =middle, xlabel=$x$, ylabel=$y$,
            every axis x label/.style={at=(current axis.right of origin),anchor=west}
          ]

       \pgfplotsinvokeforeach{-2,...,2}{
         \ifnum #1 = -2
           \addplot [thick, red, smooth,domain=-2:2] {x^3+#1} coordinate[pos=0.68] (n#1);
         \else
           \addplot [blue, smooth,domain=-2:2] {x^3+#1} coordinate[pos=0.64-0.02*#1] (n#1);
         \fi
         \node[inner sep=1pt,pin={[anchor=west]30:$C=#1$}] at (n#1) {};
        } 
        \addplot[soldot,red]coordinates {(1,-1)} node [anchor=west,text=black]  {$(1,-1)$};
        \node at (axis cs:-0.6,-0.3) [anchor=west] {$0$}; 
   \end{axis}
\end{tikzpicture}
\end{document}

\documentclass[pstricks]{standalone}
\usepackage{kpfonts}
\usepackage{pst-plot,multido}
\begin{document}

\begin{pspicture}(-2,-5)(3.5,5)
\psaxes[labels=y]{->}(0,0)(-2,-5)(2,5)
\multido{\iA=-2+1,\rA=1.1+0.05}{5}{%
  \psplot[linecolor=\ifnum\iA>-2 blue\else red\fi,algebraic,linewidth=1.5pt]{-1.5}{2}{x^3+\iA}%
  \psline(*\rA\space x^3+\iA)(!2 \rA\space 3 exp \iA\space add)
  \uput[0](!2 \rA\space 3 exp \iA\space add){$C=\iA$}}
  \psdot[linecolor=red,dotscale=1.5](1,-1)
  \uput[-45](1,-1){$(1,-1)$}
\end{pspicture}

\end{document}

A way to do this with MetaPost, for whom it may interest. The problematic pointing segments are build thanks to the handy cutafter macro. As its names suggests, it doesn't draw the part of a given path located after its intersection with another given path.

The MetaPost code is included in a LuaLaTeX program for convenience.

\documentclass[border=3mm]{standalone}
\usepackage{luatex85, luamplib}
    \mplibsetformat{metafun}
    \mplibtextextlabel{enable}
\begin{document}
\begin{mplibcode}
picture curves; path graph[];
u := cm; xmin = -2.5; xmax = 3; xstep := .1; ymax = -ymin = 4;
beginfig(1); 
  % Axes
  drawarrow (xmin*u, 0) -- (xmax*u, 0);
  label.bot("$x$", (xmax*u, 0));
  drawarrow (0, ymin*u) -- (0, ymax*u);
  label.ulft("$y$", (0, ymax*u)); 
  % Clipped cubic curves
  curves = image(
    for C = -2 upto 2:
      graph[C] = function(2, "x", "x**3+(" & decimal C & ")", xmin, xmax, xstep) scaled u;
      draw graph[C] withcolor if C = -2: red else: blue fi;
      label.ulft("$" & decimal C & "$", (0, C*u));
    endfor;);
  clip curves to 
    unitsquare xyscaled (xmax-xmin, ymax-ymin) shifted (xmin, ymin) scaled u;
  draw curves;
  % Equation, and label for (1, 1)
  z = (u, -u); drawdot z withpen pencircle scaled 3bp withcolor red;
  label.rt("$(1, -1)$", z); label.rt("$y = x^3 + C$", u*(xmin, .75ymax));
  % Labels C and their segments
  for C = -2 upto 2:
    pair loc; loc = u*(1.75+.06(C+2), 2.5+.6C);
    label.rt("$C = " & decimal C & "$", loc);
    draw (loc -- loc+infinity*dir (-165)) cutafter graph[C] ;
  endfor;
endfig;
\end{mplibcode}
\end{document}