The best analogy I can think of is that of hiring a mechanic.
Without dependency injection, you hire a mechanic and the mechanic brings his own tools. He may have lousy tools, he may have great tools, he may be using a pipe wrench when he should be using a socket. You don't know, and may not care, so long as he gets the work done.
With dependency injection, you hire a mechanic and you provide him with the tools that you want him to do his work with. You get to choose what you consider to be the best or most appropriate tools for the work you are hiring him to do.