To make sure this makes sense, let's first look at the concept of empathy - to some it may carry connotations of peace, love & flowers but what is it?
Well, empathy is "the capacity to recognize emotions being experienced by another sentient [...] being", as put by our friend Wikipedia - quite simply, it's the ability to put yourself in someone else's shoes.
This is both natural and tricky at the same time. On one hand, it's known by even the hard sciences that human beings are, in fact, hardwired to empathize with one another - and yet, on the other hand, we also know that we are intrinsically subjective; we cannot see except through our own eyes.
Basically, empathy is just like other normal functions - let's say, for example, running: Everyone can do it at a basic level but some do it better, and that's because they've trained themselves, and specialized; you can't both sprint well and be a good marathon runner.
Likewise, as a C/UX specialist, it isn't just about a general empathy for your fellow man but about having honed your skill with this particular human function into a tool that doesn't writhe in your hands.
The reason this is such an important ability is that, looking at products and services, measuring them, analyzing them and trying to determine their effect and effectiveness, the interpretation of all this data requires the additional understanding that these users, these people, are all different - from each other and from one self.
We don't need to know what I'd do in their place - we need to know what they're actually doing. Or wanting, or liking, or disliking, buying, skipping, recommending or berating.
To understand why this is the most important tool, we should also look at a few others at our disposal, so let's do that:
Data fields - I mentioned this in my article about Big Data vs. UX; data is neutral (some would even say neutered). We can measure how many times a video is shared, or how many people download an app or retweet a tweet but that data doesn't tell us anything else. Did people share the vid because it's great, or terrible - was it a recommendation, or a warning? Do they use that app, and if not, why? Why did they download it? Do they agree or disagree with the tweet, do they even think about that at all? And so on.
Personas - this is a very popular way of trying to formalize empathy, by ascribing particular qualities to a fictional person, and it can indeed be useful but there's a little known (or considered) pitfall, and it's the same one faces when dealing with statistics: A persona isn't an actual person, any more than a statistical average is - it's a construct, and as such behaves in a way that's radically different from how actual, real people behave.
You can't sell a product to a statistical average.
Questionnaires - also often used, and often useful, but again, there's a caveat, and it's almost and addage around us UX folks: What people say and what they do are two very different things. Not to mention what people say about what they do; we're very good at coming up with an explanation for our own behavior that suits or self-image.
For each of these tools, the difference between "useful" and "less useful to potentially misleading" lies in the application of empathy on the interpretation - the understanding that, in the end, we are dealing with actual people, complex, confusing, and wonderfully different.
So there you go - that's why it is so important. Any questions...?