For me, being an enthusiastic teacher means greeting everyone with a smile and making every interaction (with both students and parents) as positive of an experience as possible. In the lesson room, enthusiasm translates to investment. Being invested in your teaching means caring about the students' goals and finding out what makes them tick. In what format do they learn most effectively? What inspires them musically? In your spare time, listen to some of your students' favorite bands/artists. Even if it is drastically different from what you would choose to listen to, you may gain insight on where they are coming from (and who knows, you might just like it!).
Being enthusiastic with our encouragement is also key to students success (particularly if they are in a pivotal popint in their development). Acknowledge areas where they are lacking, and work towards fixing them; but always emphasize the positive and their successes. When correcting improper playing/technique, always explain the "why" and what they will gain from making the proposed changes. Explaining that making the changes wil allow them to play more like their favorite artist(s) is always helpful in making a connection.
Additionally, having enthusiasm for your students' goals is very important. Being invested in your students' goals means supporting them in their playing style/genre of their choice. Forcing students to play in our own preffered genre often backfires and students frequently lost the motivation that first caused them to enroll in lessons. While it is necessary to expose students to different genre and for them to be a versatile player, I never force a single style of music/playing upon them. In my teaching, I have found it most effective to start with a genre-neutral curriculum which allows students to then branch off into their chosen style(s). In the beginning, it is all about making connections. While many teenage rock students may balk at playing "Yankee Doodle" when learning to read music, it's important to explain that developing their reading is a necessary first-step in working towards playing the rock anthem they have their heart set on. Over time, I have found it most effective to divide each lesson into two parts: the first half being "book work"/technique, and the second half spent applying those techniques to a song we chose together.
As teachers, we all have different techniques for inspiring our students, but I have found my enthusiasm and invested style of teaching to be particuarly effective (and directly linked with student success). It is all too easy to let our teaching take over and overshadow the student. Remembering to listen to the student is essential for their development (and ours as teachers). Above all share your passion for music and remember to foster it in others. Your students will thank you for it.