Why does the shape of a neuron suit its job?
The purpose of a neuron is to communicate between cells. Therefore we can assume that a cell whose primary purpose is communication will need to be able to communicate rapidly, with many other cells, over long distances, and do so with high fidelity and low downtime between signals. We can easily visualize circumstances where all these conditions would be required, such as the sight of an oncoming car triggering muscles in the legs to move.
The general shape of a neuron is often compared to a tree. On one side of the familiar nucleated cell body is a system of complex, branching extensions of the membrane called dendrites. The dendrites look very similar to tree roots and allow the neuron to cover a large area, increasing the chances of it communicating with many other cells via connections between the dendrites and the other cells. Neurons often connect to thousands of other cells in the brain. The shape of the dendrites, by "flowing" into the cell body, allow the signals that the neuron receives to be gathered, evaluated and acted upon by the cell's central control; i.e. it can decide whether to transmit the signal or ignore it.
After the cell body is the axon, which is a long (relatively speaking, but some can be over 1 meter long) tubular extension of the membrane, often covered periodically by other cells that form what is called a myelin sheath. This portion of the neuron is often compared to an electrical wire; the axon transmits electrical signals, and the surrounding myelin insulates it, allowing the signal to travel farther and more efficiently.
At the end of the axon is another series of branches, though not so numerous as the dendrites. These branches, the presynaptic terminals, connect to other neurons and utilize the axon's electrical signals to release neurotransmitters to those other cells.
Overall, the neuron's shape allows it to be a good "listener" because of its numerous and widespread dendrites, and efficient transmitter because of its centralized cell body and long, insulated axon.