So I understand that when dy/dt is $0$, this means that $y(t)$ is a constant. But why does the number of limiting behavior depend on the roots of the differential equation. And why if $dy/dt ≠ 0$, the solution curves tend to approach or avoid the $dy/dt = 0$ solutions.
For example, the graph below is the directional field of y vs t of the equation ${dy\over dt} = y^2 - ty.$ As you can see the surround solutions seem to approach 0 or t which are roots for dy/dt = $0$.
They teach us how to calculate this equations but they don't truly teach us the meaning behind these calculations. So please excuse me if this seems trivial.
I think what you mean is something like this. Consider a first-order autonomous differential equation $$\dfrac{dy}{dt} = f(y)$$ where $f$ is a continuous function. If you have a solution $y = s(t)$ with $\lim_{t \to \infty} s(t) = L$, then $L$ must satisfy $f(L) = 0$. This is easy to prove by contradiction: if $f(L) \ne 0$ then there are $\delta > 0$ and $\epsilon > 0$ such that either $f(y) > \epsilon$ for $L-\delta < y < L + \delta$ or $f(y) < -\epsilon$ on that interval.
Any solution that ever has $L - \delta < y < L+\delta$ will then exit that interval in time at most $2\delta/\epsilon$ and can never return...
EDIT: For a non-autonomous equation $\dfrac{dy}{dt} = f(y,t)$, all sorts of other behaviours are possible. The isoclines $f(y,t) = 0$ are important, but usually they do not represent asymptotes of the solutions. In your particular example you are correct, there does happen to be a solution that approaches $y=t$ asymptotically as $t \to +\infty$. It is the separatrix between the solutions that go to $+\infty$ in finite time and those that go to $0$ as $t \to \infty$.
EDIT: Hubbard's terminology is useful: a curve $y = g(t)$ is a lower fence if $f(g(t),t) > g'(t)$ and an upper fence if $f(g(t),t) < g'(t)$. Thus a solution curve can cross a lower fence from below to above but not from above to below, and it can cross an upper fence from above to below but not from below to above. A funnel is a region above a lower fence and below an upper fence; solution curves can enter a funnel, but can't leave it. An antifunnel is a region above an upper fence and below a lower fence: a solution can leave a funnel, but can't enter it. However, an antifunnel always contains at least one solution curve that stays within the antifunnel.
In your example, $y = t$ for $t > 0$ is an upper fence, while for any $\epsilon > 0$, $y = t + \epsilon$ for $t > 1/\epsilon$ is a lower fence; there is one solution curve that stays in the antifunnel between these fences.