So, one can read bytecode to implement an interpreter / JVM. One can write / generate bytecode when implementing a Java compiler or a compiler for another language that will target the JVM (e.g. Scala and Jython). You might perform bytecode manipulation to optimize bytecode (if you want to produce and market a bytecode optimizer or you need it as an internal tool to give your company's code an edge over the competition). In a similar vein, you might manipulate bytecode in order to obfuscate it prior to distribution. You might also perform bytecode manipulation for aspect-oriented programming; for example, you might want to insert hooks (maybe for timing or logging purposes or for some other reason), and if it were simpler or less expensive to manipulate the bytecode than to edit all the source files (such as might be the case if the source code is unavailable or from many different sources, not all of which may be under one's control or for which it might be expensive and time-consuming to convince those teams to add such hooks), this might be a case where it would make sense to insert the modifications to the final bytecode output rather than to attempt to modify the original code (which might require upstreaming or maintaining a separate fork, or purchasing the source code from a third party that supplies only the bytecode).
You can manipulate bytecode yourself, although there are many existing open source libraries and frameworks to do it, including BCEL and ASM to name just a couple.