Empathy, defined as the ability to understand and share others' perspectives and emotions, is essential in software engineering (SE), where developers often collaborate with diverse stakeholders. It is also considered as a vital competency in many professional fields such as medicine, healthcare, nursing, animal science, education, marketing, and project management. Despite its importance, empathy remains under-researched in SE. To further explore this, we conducted a socio-technical grounded theory (STGT) study through in-depth semi-structured interviews with 22 software developers and stakeholders. Our study explored the role of empathy in SE and how SE activities and processes can be improved by considering empathy. Through applying the systematic steps of STGT data analysis and theory development, we developed a theory that explains the role of empathy in SE. Our theory details the contexts in which empathy arises, the conditions that shape it, the causes and consequences of its presence and absence. We also identified contingencies for enhancing empathy or overcoming barriers to its expression. Our findings provide practical implications for SE practitioners and researchers, offering a deeper understanding of how to effectively integrate empathy into SE processes.