Neutral objects can be attracted to charged objects through a process called induction. This works because all matter is composed of positive protons and negative electrons. While the protons are basically fixed in place, the electrons are able to move somewhat within a substance. Remember also that like charges repel and opposite charges attract
In your example when a negative charge is brought near a neutral object the negative electrons in the neutral object will be repelled. They will try to move away from the approaching negative charge. When they do this, the neutral object appears to have a small positive charge on the side where the negative charge is approaching. A temporary charge separation has been "induced" in the neutral object, egen though its overall charge is still neutral. If the negative charge is then removed, the electrons in the neutral object will move back to their original positions. Since opposite charges attract, the neutral object will now move toward the negative charge.
Now consider what will happen when the negative charge is brought in contact with the neutral object. As the negative charge approaches, as explained above, there is a positive charge on that side of the neutral object. When the two objects make actual contact, some of the electrons from the negatively charged object will actually move to the neutral object to try to neutralize the apparent positive charge. Now when the objects are separated both objects have a negative charge.