To empathize with someone means to:

Empathizing with someone involves the ability to understand their state of mind and to emotionally connect with their experiences. This means you are able to recognize and appreciate what the other person is feeling, which allows for a deeper level of communication and support. When you empathize, you step into their shoes and perceive their emotions from their perspective, fostering compassion and understanding.

The other choices do not align with the definition of empathy. Persuading someone to your viewpoint implies a lack of understanding for their feelings and experiences, as it focuses on convincing rather than connecting. Ignoring someone's feelings and dismissing their concerns are actions that run counter to the essence of empathy, as they involve a disconnect and a lack of recognition of the other person's emotional state. Thus, understanding their state of mind is the fundamental aspect of empathy.