The point is that while Newton's third law always produces a counter force, this counter force does not act in the same particle/object.
If you punch me in the face with some punching force, then my face will in response exert that same force to your hand. But my face will still be jerked backwards. It only feels the punching force, while it doesn't feel the counter force. Only your hand feels the counter force. The punching force on my face is not balanced, but results in a net force backwards which causes an acceleration backwards (according to Newton's 2ns law). The counter force is at the same time exerted on your hand which is why your hand is slowed down at impact.
In a spring that same principle is repeated in each connected particle.
- You pull in the particle at the spring end. A force is exerted on it outwards. To start with this force is not balanced, so the particle moves.
- As it moves a bit, Hooke's Law tells us that a spring force is created between the particle and its neighbour due to its elongation of the bond. So a spring force is exerted in the neighbour particle which then in turn starts to move.
- This particle in turn pulls backwards in the first particle due to Newton's third law, so you feel sine resistance to your pull. But your pull is still bigger, so the resulting force is still outwards and the particle you are pulling in still moves.
- This neighbour pulls in its neighbour, which pulls in the next neighbour etc. The force propagates all the way to the final particle at the other end.
- If this particle at the other end is fixed (maybe glued to a surface), then it can't move. So the elongation grows between the last and the second-last particle. This increases the spring force. The second-last particle in turn pulls a bit more in the third-last particle and increases the elongation there as well so the force grows. Etc. This propagates from particle to particle all the way back to your hand.
The longer the elongation, the larger the spring force between them. At some point the spring force on the first particle balances out your pulling force, and you can't elongate it any further (no net force). Then the spring is stretched as much as it can.
It took some time for this counter force to grow large enough to balance out the pull, because the response had to propagate from the first to the last to the first particle again. This propagation is delayed a bit (takes a bit longer) in elastic materials.
If you pulled in a very rigid object (a spring made of glass or granite for example), then the force you exert in the first particle is immediately propagated to the end and the response propagated back to the first particle again. The pull is therefor immideately balanced out and no extension happens. This is what we call a very stiff and strong material.