This might seem like an inappropriate question and even though its easy to retort using analytic geometry, I don't feel convinced that a tangent really does intersect a circle at only one point. Maybe it's better to ask why does it look like that the tangent intersects the circle at more than one point?
If, for example, we have a circle with the equation $x^2 + y^2 = 100$, and a tangent to this circle $y = 10$, then although substituting the value of $y$ in equation of the circle gives just one answer in the form of $x^2 = 0$, but when I look at the diagram of these equations and think about how if we place an extremely large straight bar on a (large enough) circular body, the bar won't really touch the circular body at just one point. I was also going to write about celestial bodies but I changed my decision because I realized that they aren't perfectly circular (or spherical).
How can I convince myself that a tangent intersects a circle at just one point?