Why is it important to study the different periods and performance venues in the arts?

Artists influence each other. Within a particular context, such as Paris during the early modernist period, artists of a generation often have contact with each other (or at least each others' works); this works across artistic media as well as within a particular medium. For example, Gertrude Stein spent a lot of time in Picasso's studio, sitting for the portrait he made of her shortly before he developed cubism. While we will never know the content of their conversations, Stein's writing changed at this point as did Picasso's painting. 

Not only do artists of a generation influence each other, but artists are also influenced by previous generations, whether in carrying on certain traditions or choosing to rebel against a set of traditions. In order to understand Brahms' symphonies, we need to understand Bach because Brahms very much appreciated Bach's approach to composition and used some of his techniques. In order to understand Bach, we need to be aware of Renaissance music and the limitations of its approach to acoustics that Bach addressed in his series of pieces called "The Well-Tempered Clavier." 

Contexts are critical, too. For example, prior to the invention of the camera, one of the main purposes of painting was to preserve an image, whether of a person, a landscape, and so forth. Painters worked on their ability to depict a scene with accuracy. Following the invention of the camera, painting was freed from the need to be visually accurate and painters became more able to explore all the possibilities of the medium. It is no coincidence that modernism arose so shortly after the camera. Painters such as Matisse, Picasso, and Duchamp began to explore the limits of representation and other painters such as Kandinsky discarded representation for the creation of abstract works. 

In short, artists "stand on the shoulders of giants," to quote Isaac Newton; in order to understand their work, artists need to understand the giants. In order to participate in the art world, as well, aspiring artists need to understand why things are as they are so they can make choices about their contributions to this world. And, in order to appreciate art fully, aficionados need also to have knowledge about art history. The shorter answer to this question, and probably the best one, is that learning about the history of various art forms is inherently interesting because artists' lives have as much to teach us about being human as their works.