Zig Ziglar said that before people care about you, you must care about them. Emotional connection comes first so don't try to initially sell people on your skillset (logic).
With empathy, you'll be able to avoid miscommunications and overcome conflicts. Even if you're not good in the first place, trust me when I tell you people will stick with you for the long term. They'll come along for the ride.
Just remember that empathy is a skill which is totally learnable. Study people's communication styles and body language. People watch when you're out. Over time you'll begin to see patterns and gain experience. This 1 skill can be the key difference between success and failure in your piano teaching career.