The diagram tells you that the force on a positive charge will be along a tangent to the electric field line at point $A$ and will be directed in the same sense as the arrow on the electric field line.
This gives you the direction of the acceleration and hence the rate of change of velocity of the charge.
If at point $A$ the charge is at rest then the charge will start to move along the tangent.
However in general although you know the direction of the force on the test charge, its acceleration and its rate of change of velocity unless you know the velocity of the charge you cannot predict which way it will continue to move.
That being said that even though you can predict the direction in which a stationary charge will start to move it will not continue moving along the electric field line unless the electric field is uniform.