Indirect interactions are interactions wherein one species affects the fitness of another species through their direct interaction with a third intermediary species which will remain unaffected. For example, in a community with species A, B, and C, species A affects species B which will in turn affect species C. So species A does not directly affect species C, only indirectly. Furthermore, the chains of indirect interactions observed in nature rarely exceed 5-6 species (see source below).
For apparent competition, a shared predator or herbivore facilitates the competition between two competitors. With my previous example, species A and C are competitors whose interactions are affected by their predator species B. Without species B, there will be increased competition between species A and C, which can lead to a decrease in diversity in the long run. I'm not sure what you mean by apparent mutualism but another example of indirect interaction is indirect commensalism wherein a species benefits from another species indirectly through the third intermediary species, which does not gain anything or is not affected.
Thus, these indirect interactions are essential in communities because they are used in determining community structure and dynamics. To some extent, the stability of a community may be affected if the intermediary species is removed.
I also attached some sources you could refer to. Hope this helps!