Great responses already in this thread. One more perspective worth adding:
Empathy is the ability to think and feel the way someone else does. It's different from sympathy or caring, which are more about how you think about someone else.
Empathy is important in product discovery because great product discovery is less about your product or feature -- which may not even exist yet -- and more about understanding your target customer. Great product discovery is about learning what's important to them, in a specific context that they care about. Empathy helps you gain insights into what they value most in that context, so that you can create products, features and experiences that your target customers will value.
Your ability to design products that your customers will truly value is directly proportional to your ability to think and feel about that what's valuable in that context in the same ways that they do. This is empathy.
In order to gain empathy, you need to understand how others think, and be able to adopt their own values, mindset, preferences, feelings, etc. This means leaving your own biases and assumptions behind, no matter how much of a product expert you are. This kind of insight can only come from direct interaction -- empathy can't be gained by thinking about a problem, collecting survey data, or studying product analytics.
Fortunately, empathy is a skill that can be learned. Like any skill, it benefits from good guidance or training and improves with practice.