There is a logical reason to be compassionate, to make others feel better which benefits them. Especially if the person is somebody you know. Making others feel better is said to be a cure for depression, which is another reason it's logical to be compassionate towards other people. Conclusion: being compassionate and empathetic benefits both yourself and the person you express it towards.
We can get further in the world as a group rather than as individuals, so helping those around us will benefit us. Compassionate and empathy help those we depend on and indirectly help us. I'm sure there are other reasons, but this seems to be a core from my perspective.
Humanity as a whole are predators by our very nature. It is embedded into our very genetic makeup to attack anything weaker than us. If you compare homo sapiens to any species of carnivorous predators the all instinctively governed by the same basic principle. Only the strong survive. Emotions do not influence their actions as they do most humans. They're true strength lies in their ability lack of compassion and sympathy. Anything that may weaken the unit is eliminated, from sick or injured pack members to weak offspring. Humanity's tendency towards being controlled by emotions weakens our natural desire for self preservation. This is a deadly vulnerability not found anywhere else in the animal kingdom, it will in truth likely be the cause of our ultimate destruction.