A human (or almost any other animal) would eat a cup of yogurt by ingesting it through their mouth. However, fungi don't have mouths, nor do they have arms or legs to move around or get the yogurt to them. So what do they do?
Wherever the fungi is, they simply absorb everything around them that they can. Instead of reaching for the yogurt, a fungi spore would end up in the yogurt cup randomly, and just absorb all the nutrients from the yogurt passively.
Fungi cells have an extremely efficient surface area to volume ratio, which means that each cell has a ton of surface area with which to absorb nutrients. Once they absorb it through a process not unlike endocytosis, they break down most of the nutrients they absorb using various enzymes. Hydrolytic enzymes break down polysaccharides, proteins, and lipids, cellulase breaks down cellulose.
The part that gets complex is that there are millions of types of fungi. Some are parasites and leech nutrients off of their hosts, others are symbionts that help provide nutrients to their hosts (for example, some fungi actually live in the roots of a plant and help break down nutrients for the plant to absorb), and others are decomposers (like mushrooms). Each one has a slightly different method of absorbing nutrients and a different source, but the basic enzymes and processes are the same.