Excellent question!

One theory is the empathy-altruism thesis, which argues that empathy indeed leads to genuinely altruistic, other-centered motivation. Genuinely altruistic motives are taken into account by the individual agent in deliberating about whether or not to help. The question of whether the agent will act on his or her altruistic motivations depends ultimately on how strong they are and what costs the agent would incur in helping another person. C. D. Batson has conducted many experiments to try to prove this thesis, and it is the thesis toward which I am inclined.

Another theory is the egoistic interpretation of empathy-related phenomena, where empathizing with other persons in need can lead to a heightened awareness of the negative consequences of not helping, such as feelings of guilt, shame, or social sanctions. Alternatively, it can lead to an enhanced recognition of the positive consequences of helping behavior, such as social rewards or good feelings. According to this interpretation, we help others only because we recognize helping behavior as a means to egoistic ends. It allows us to reduce our negative feelings, to avoid "punishment," or to gain specific internal or external "rewards."

From the embodied/enactive perspective, which draws on biology, phenomenology, and cognitive science, empathy is an essential part of our interactions with people, other living beings, and cultural artifacts, including aesthetic creations (visual arts, music, dance, literature, etc.). Here is some pertinent literature:

Brinck, Ingar

2007 “Situated cognition, dynamic systems, and art: on artistic creativity and aesthetic experience,” in Janus Head (Trivium Publications), vol. 9, no. 2, pp. 407-431 (http://www.janushead.org/9-2/Brinck.pdf, accessed: 5 March 2016).

2017 “Empathy, engagement, entrainment: the interaction dynamics of aesthetic experience,” in Cognitive processing (Springer) (http://link.springer.com/article/10.1007/s10339-017-0805-x, uploaded: 8 April 2017, accessed: 18 April 2017).

Burke, Michael; Kuzmičová, Anežka; Mangen, Anne; Schilhab, Theresa

2015 “Empathy at the confluence of neuroscience and empirical literary studies,” in Scientific Study of Literature (John Benjamiins Publishing Company), vol. 6, no. 1, pp. 6-41 http://journal.frontiersin.org/article/10.3389/fnhum.2015.00442/full, accessed: 4 April 2017).

Decety, Jean

2011 “The neuroevolution of empathy,” in Annals of the New York Academy of Sciences (New York Academy of Sciences/Wiley Periodicals), vol. 1231, pp. 35-45 (http://onlinelibrary.wiley.com/wol1/doi/10.1111/j.1749-6632.2011.06027.x/full, accessed: 18 April 2017).

Foster, Susan Leigh

2005 “Choreographing empathy,” in Topoi: An International Journal of Philosophy (Springer), vol. 24, no. 1, pp. 81-91 (http://link.springer.com/journal/11245/24/1/page/1, accessed: 8 January 2017).

2009 “Movement’s contagion: the kinesthetic impact of performance,” in The Cambridge companion to performance studies, Tracy C. David, editor, Cambridge, Cambridge University Press, pp. 46-58 (http://www.yavanika.org/classes/reader/fosterkinaesthetic.pdf, accessed: 26 December 2016).

Freedberg, David; Gallese, Vittorio

2007 “Motion, emotion and empathy in esthetic experience,” in Trends in Cognitive Sciences (Cell Press/Elsevier), vol. 11, no. 5, pp. 197-203 (http://www.sciencedirect.com/science/journal/13646613/11/5, uploaded: 7 March 2007, accessed: 30 January 2015).

Gallese, Vittorio

2001 “The ‘shared manifold’ hypothesis, from mirror neurons to empathy,” in Journal of Consciousness Studies (Imprint Academic), vol. 8, no. 5/7, pp. 33-50 (https://www.researchgate.net/publication/287856543, accessed: 30 September 2016).

2005 “Embodied simulation: from neurons to phenomenal experience,” in Phenomenology and the Cognitive Sciences (Springer), vol. 4, no. 4, pp. 23-48 (https://www.researchgate.net/publication/225213240, accessed: 22 November 2015).

2014 “Bodily selves in relation: embodied simulation as second-person perspective on intersubjectivity,” in Philosophical Transactions of The Royal Society B: Biological Sciences (The Royal Society), vol. 370, no. 1663 (http://rstb.royalsocietypublishing.org/content/369/1644/20130177.full, uploaded: 28 April 2014, accessed: 11 January 2017).

2016 “Finding the body in the brain, from simulation theory to embodied simulation” (preliminary version), in Goldman and his critics, Brian P. McLaughlin and Hilary Kornblith, editors, Chichester/Malden/Oxford, John Wiley and Sons, pp. 297-317 (https://www.researchgate.net/publication/300056992, accessed: 24 April 2017).

Gallese, Vittorio; Schneemann, Carolee

2017 “In conversation” (preliminary version), in Mirror touch synaesthesia, threshholds of empathy with art, Daria Martin, editor, Oxford/New York, Oxford University Press, pp. 39-54 (https://www.researchgate.net/publication/318081499, uploaded: 1 July 2017, accessed: 1 July 2017).

Gangopadhyay, Nivedita

2014 “Introduction: embodiment and empathy, current debates in social cognition,” in Topoi: An International Journal of Philosophy (Springer), vol. 33, no. 1, pp. 117-127 (https://link.springer.com/article/10.1007/s11245-013-9199-2, uploaded: 13 October 2013, accessed: 3 June 2017).

Gerger, Gernot; Pelowski, Matthew; Leder, Helmut

2017 “Empathy, Einfühlung, and aesthetic experience: the effect of emotion contagion on appreciation of representational and abstract art using fEMG and SCR” (preliminary version), in Cognitive Processing (Springer) (https://www.researchgate.net/publication/314504129, uploaded: January 2017, accessed: 10 March 2017).

Kesner, Ladislav; Horáček, Jiří

2017 “Empathy-related responses to depicted people in art works,” in Frontiers in Psychology (Frontiers Media), vol. 8, article 228 (http://journal.frontiersin.org/article/10.3389/fpsyg.2017.00228/full, uploaded: 24 February 2017, accessed: 18 April 2017).

Rizzolatti, Giacomo; Craighero, Laila

2005 “Mirror neuron: a neurological approach to empathy,” in Neurobiology of human values, Jean-Paul Changeux, Antonio R. Damasio, Wolf Singer, and Yves Christen, editors, Heidelberg, Springer-Verlag, pp. 107-123.

Zahavi, Dan

2017 “Phenomenology, empathy, and mindreading,” in The Routledge handbook of philosophy of empathy, Heidi L. Maibom, editor, London, Routledge, pp. 33-43 (https://www.academia.edu/33842755, accessed: 16 July 2017).

One of the very best current philosophers to address this question in the phenomenological tradition is Dan Zahavi. Type in "Dan Zahavi empathy" in Researchgate and you will find multiple contributions. He is working from the basis laid down in Husserl's work--especially Cartesian Meditations and vols 13-15 of Husserl's collected works (in German, "Husserliana"). Edith Stein's On the Problem of Empathy is one of the most important early works building upon Husserl. For Husserl and Stein, empathy is not an inference but rather a kind of perception. But the sense in which it is a perception is highly problematic. Empathy is also a key concept in Alfred North Whitehead's philosophy. For Whitehead, emotional empathy is nothing less than the essence of the mutual immanence of all "occasions of experience" (and occasion of experience is nothing less than the fundamental category of reality, comprising both what is normally considered mental and physical). Also, for Whitehead, empathy is the essence of art. "The origin of art lies in the craving for re-enaction. In some mode of repetition we need by our personal actions, or perceptions, to dramatize the past and the future, so as to relive the emotional lives of ourselves and our ancestors." Adventures of Ideas (New York: Macmillan, 1971), p. 271. I am not very familiar with contemporary process thought, but I would be surprised if empathy does not play a key role.