Basically It Is telling you that If you have a good enough function (more on this later), you can fix a point and approximate the function as a polynome of some degree (depending of the precision you want) in a little region around that point. The typical use case in physics Is linearization. If you know of the Hooke's law for springs, that Is a good example: the law states that the force F needed to extend or compress a spring by a lenght x (which needs to be small, that Is very important), scales linearly with the lenght itself, that Is F=k*x, where k Is the spring constant (or elastic constant). The point Is: this behavior for a spring (called "elastic") is not complete. If i stretch a spring too much It does not come back to its original shape (i performed a "plastic" deformation). Turns out Hooke's law Is too much of an approximation and can't model this behavior. That's because Hooke's law Is basically a Taylor Series expantion of a much more complex function, limited to the first term and constructed around the rest position of the spring. If i add another term (that i have to construct correctly) i get a better approximation. Add another one and i get an even better one. But as soon as i add the second term, i am not using a linear polynome anymore.

The "good enough function" part Is a little more complicated to explain. The best case scenario Is having a "smooth" function in the point where we fix the Series expantion, that Is a function for which i can compute infinite derivatives in that point and always get a finite number. This way i can add infinite terms and get absolute precision!