Research is a conversation among a community of researchers. To join the conversation, you need to catch up on what has been said, what people are interested in, and have something worth saying. Otherwise people will simply ignore you and keep talking to each other.
The undergrad curriculum catches you up to the conversation c. 1900. You start to speak the language that the researchers are using.
A graduate curriculum gives you the basics of what the researchers are talking about in many different conversations, and often brings you up to speed to the conversation c. mid/late 20thc - when many of the senior people in the field were just starting.
You pick a conversation. You study a field in depth and read papers and preprints until you're caught up to the conversation in broad strokes. Ideally you have a mentor or advisor, who helps teach you what is interesting and introduces you to other researchers and can vouch for your potential.
You meet many of those people in person. You hear how they talk about their subjects among themselves, you learn the tricks they use to think about the subject efficiently.
As you immerse yourself in the conversation, you become aware of the problems that are interesting to other researchers, and how you might take your knowledge to solve them. After working hard, you answer one of these questions, and you start talking about your answer by email - in person - in talks - via the arxiv - eventually publishing.
Because you have said something interesting, the other researchers become interested in what you have to say, and you are on your way to joining the conversation.
Neal