I keep hearing that I should be networking my way into my next job. What does that mean?
You're networking now!
"Networking" is a shorthand way to say "talk with and listen to other people."
Many jobs are never advertised. Employers fill them with people they know, or with people that trusted friends and acquaintances recommend. It's very expensive and time-consuming to hire people, so naturally they want to hire a low-risk candidate. If a current employee recommends someone, that person is much more likely to be hired.
That's where networking comes in. If someone you know can recommend you for that dream job, you are already halfway there. The trouble is that you may not know the person who can make that recommendation. So you'll want to expand your circle of acquaintances.
There are lots of ways to do that. Talk with anyone you can, and be sure to listen too. Schedule an informational interview with someone who does what you want to do. Chat with people at the library, the gym, the doctor's office, your church/synagogue/mosque. Get to know people who know a lot of people: the person who cuts your hair, the clerk at the grocery store, American Job Center staff, etc.
The more people you get to know, the more likely you will stumble upon the one person who knows what you need to know: a job opening, the name of someone who can be helpful, a particular program, a company that might interest you. This is hard if you're shy; but remember that most people like to help people. They can't help if they don't know what you need.