Conceptually the reason why the field lines point inward is that gravitational fields are potential for a force. A single mass creates field two masses create a force. The potential force (gravitational field) should be in the direction of the attractive force of gravity, if you drop a ball above the Earth (regardless of the height) it is pulled toward the Earth.

Sir Isaac Newton's law of universal gravitation (i.e. the law of gravity) can be restated into the form of a gravitational field. Instead of calculating the forces between two objects every time, we instead say that an object with mass creates a gravitational field around it. The gravitational field is defined as the force of gravity at a given point divided by the mass of an object at that point.

This equation depicts a vector field around M which is always directed toward it, with a value equal to an object's gravitational acceleration within the field. The units of the gravitational field are m/s^2.

Because of the vector nature of gravity, the gravitational field created by a mass is a vector field. Like all vectors, vector fields are also additive. Mapping a single mass we get a vector field we get the following image.

Remember that the strength of the gravitational field is proportional to 1/r^2. That means if we double the distance away from a mass then the strength of the field drops by a 1/4, triple the distance the field strength is 1/9 it's original strength.

Then when another mass is introduced, the fields created from the new mass interacts with the old field. The field in between the points gets weaker, and a certain point the field goes to zero. While in the space beyond the two points the field gets stronger. Introducing a third mass, the regions between the masses the field gets weaker even still, outside slightly stronger.