How do I visualize differential equations?

For $y^{\prime} =y+1$, the slope of a solution must be negative at any point where $y<-1$ and positive otherwise. So in your pictures, this eliminates the straight line and the parabola.

For $y^{\prime} = -2xy^2$, note that $y^2$ is non-negative, so $-2xy^2$ is negative in if $x>0$ and positive if $x<0$. Only one of the pictures shows a solution which is increasing for $x<0$ and decreasing for $x>0$.

For these sorts of problems, there's probably no single technique, but in general, think about what slope of the solution is in each quadrant. If $y^{\prime} = f(x,y)$, it might be helpful to sketch the curve $f(x,y) =0$ and then think about what happens on each side of that curve. It could help you eliminate a picture or two.

A slope field for the differential equation $$ \frac{dy}{dx} = F(x, y) \tag{1} $$ is a diagram having line segments of slope $F(x, y)$ placed at $(x, y)$, usually for points lying in a rectangular grid. Geometrically, a solution of (1) is a graph $y = f(x)$ "tangent to the field", i.e., satisfying $$ f'(x) = F\bigl(x, f(x)\bigr). $$

With a bit of practice, a function $F(x, y)$ and a plot of a slope field can be matched fairly easily; just match values of $F$ (say, points where $F = 0$) with segments having that slope.