One way to think about slope fields is to imagine them like currents on a river. If you drop a cork into the river at a certin point (the initial condition) where will the cork go? The path of the cork will always be parallel to the flow lines. On the graph I "dropped" the cork in three different place (in three colors) and sketched a rought graph of what a solution would look like. All solutions are attracted to the line y=x.

If you drop a cork at (0,0) it will stay on the flow line y=x. This represents the solution to the ODE with inital condition y(0)=0. I hope this helps.How do you interpret that graph? Is that straight line that runs through the origin part of the solution?