Showing initiative means that you begin to master your work environment and need less direction from others. You know what needs to be done and start on making those things happens. People sometimes feel that taking initiative means you should take on as many projects as possible but this isn't true. It means completing your job duties to their maximum without prompting from your boss and then moving onto new duties that indicate your quality to the workplace.
A few ways to show initiative are as follows:
Ask Questions: Ask questions about the projects you are completing.
Fulfill Your Duties Well: Make sure you complete all of your duties well.
Go Beyond Your Requirements: Fulfill your duties and go beyond the requirements to show your skill.
Help Others: When possible help others when they are behind.
Volunteer for Important Projects: When important projects arise that will showcase your abilities volunteer for them.