Previously, you have approximated curves with the graphs of Taylor polynomials.

Step 1
General in calculus the approximation of any curve is done by the lines and the graphs of the other polinomials, i.e., Taylor polinomials.
Now the circumstances in which the approximation is done using the osculating circle, observe that this is the circle that approximates the curve at a point better than all other circles.
The osculating circle of a curve at any given points is the circle that has the same tangent at that point as well as the same curvature. In the same way as the tangent line is the line best approximating a curve at any point, the osculating circle is the best circle that approximates the curve at that point.
Step 2
The osculating circle passes from three infinitesimally close points on the curve. Pick any three points on the curve and plot a circle passes through these points.
Since the points are much closed to each other, thus there is a limiting position of the circle, this is the osculating circle. The reciprocal of the radius is the curvature of the curve.
Now if the curve and the osculating circle are shown locally as graphs of smooth functions then the values of these functions. At the point of cantact first derivatives and second derivatives coincides.
Thus, as the tangent line come close to a curve at a point, the osculating circle is the best that approximates by capturing the curvature.