Final answer:
The centrosome, consisting of a pair of perpendicular centrioles, is the structure from which spindle fibers develop in animal cells. It is important for chromosome organization during cell division.
Step-by-step explanation:
The structure in animal cells that contains centrioles from which the spindle fibers develop is the centrosome. The centrosome is a microtubule-organizing center found near the nuclei of animal cells. It features a pair of centrioles, with each centriole being a cylinder composed of nine triplets of microtubules. These centrioles are arranged perpendicular to one another. During cell division, specifically mitosis, the centrosome plays a crucial role in organizing chromosomes. In contrast to animal cells, plant cells typically lack centrioles and have a structure known as the MicroTubule Organizing Center (MTOC) to fulfill a similar function for the formation of the spindle apparatus.