This is the analogue of a projectile getting launched at exactly the escape velocity, something you may remember from studying gravity in freshman physics.
Here we're talking about the photoelectric effect. The electron jumps out of the material into air or vacuum, overcoming the force of attraction that tries to keep it bound inside the material (the force of attraction comes from the surface field, the image force effect, etc.).
If the electron jumps out with too little energy, it cannot escape, but gets pulled right back in. On the opposite extreme, if the electron jumps out with much more than enough energy to escape, it will not only break free of the material but also still energy left over, i.e. it will travel away from the material with a large kinetic energy.
You are asking about the borderline case. Here the electron has just barely enough energy to escape the material, with no energy left over. So it will slow down as it moves away from the material, and get slowed down more and more as it gets farther and farther away. It will never quite come to a stop, but its velocity will approach zero.