the velocity vector field

Stand at any point in the phase plane and ask: if the system were here right now, which way and how fast would it move? The answer is an arrow. Attach that arrow to every point and you get the velocity vector field — a plane bristling with little arrows, each showing the instantaneous direction and speed of motion at its location. It is the wind map; the trajectories are the paths leaves follow as they drift along that wind.

For a planar system x' = f(x, y), y' = g(x, y), the arrow at the point (x, y) is the vector (f(x, y), g(x, y)). Its direction is where the state heads next; its length is the speed, how fast the state moves there. Where the arrow is long the system races through; where it shrinks to nothing the system nearly stops — and where it is exactly zero you have an equilibrium. A trajectory is simply a curve that is everywhere tangent to these arrows: it always goes the way the local arrow points.

The velocity field is what you can read off immediately, before solving anything — you just plug points into f and g. From it you can sketch trajectories by hand by 'following the arrows', see at a glance where motion is fast or slow, and spot equilibria as the calm points where the field vanishes. It is the bridge between the algebraic equations and the geometric picture: the equations define the field, and the field shapes every trajectory.