The inner lines and arrows of the Enneagram symbol are primarily used with the Enneagram personality types in two ways: stress/security points and directions of integration/disintegration.
From Helen Palmer's books comes the concept of stress and security points. These points are also referred to by other labels. Each Enneagram type has a single stress point and a single security point. These points are found by following the inner lines. The stress point is found by following the line moving with the arrow (e.g., the stress point for type 6 would be type 3). The security point is found by following the line moving against the arrow (e.g., the security point for type 6 would be type 9). When under stress, a type may display characteristics of it's stress point (e.g., type 6 might present type 3 characteristics when the person is under stress). Likewise, when not stressed, a type may display characteristics of it's security point (e.g., type 6 might display type 9 characteristics when the person is not under stress).
From Don Riso's books comes the concept of direction of integration and disintegration. The original concept pointed to paths of health (integration) and unhealth (disintegration). The line of integration (or health) was against the arrow and the line of disintegration (or unhealth) was with the arrow (e.g., type 6 would integrate by becoming more 9-like and disintegrate by becoming more 3-like).
More recently, the emphasis on these directions has shifted toward growth and stress. From Riso's website (the Enneagram Institute):
There are two lines connected to each type, and they connect with two other types. One line connects with a type that represents how a person of the first type behaves when they are moving toward health and growth. This is called the Direction of Integration or Growth. The other line goes to another type that represents how the person is likely to act out if they are under increased stress and pressure—when they feel they are not in control of the situation. This second line is called the Direction of Stress or Disintegration. In other words, different situations will evoke different kinds of responses from your personality. You will respond or adapt in different directions, as indicated by the lines of the Enneagram from your basic type.
Since the arrows on the inner lines can seem arbitrary, often the inner lines are thought of as simply indicating two additional influences on each personality type (e.g., a person identifying with type 6 would display both type 3 and type 9 characteristics at times independent of the ideas of stress/security or integration/disintegration).