Hi Mindy,
That drainage is called leachate. You can call it worm pee if you like but I doubt worms do pee, lol. That happens if the moisture from the worm food cannot be absorbed by the bedding (shredded newspaper and/or cardbard/corrugated cardboard) anymore. Which means the bin is going to the wet side.
For a beginner it is better not to have a bin that is too wet because there are things that can go wrong(1)or there are things that you shouldn't have done to cause this(2).

(1) the VC (vermicompost) get compacted and will not allow aeration and this in turn can cause an anaerobic condition >> stinky bin and may/will harm your worms. This is the reason some people won't use leachate. I personally use the leachate to water my outdoor plants, diluted 1:10 and in my case I haven't found any negative result. Once you are more experienced and know how to manage your bin, it is OK to have a bin that is on the wet side w/o going anaerobic. Worms DO like wet environment.
You can use a breathable material to cover your bin to help with evaporation, and aeration.

(2) this could be a result of too much food or food with too much water contents like tomatoes, melons and the like. Or not enough newspaper and the like. So, add more newspaper but DO NOT mix it with the food, just put it on top. Mixing them will cause the bin to heat up.
If you have room in your freezer, freeze the kitchen scraps in a ziplock bag or some container for a couple of days. Many people do this to kill the fruit-fly eggs that came with the veggie or fruit. Some do this so they can drain the water once defrosted.
I do not have room in my freezer and my bins are outside anyway, so I let the scraps sit for several days (pre-rotting) and drain the liquid. This is the kind of food worms like best and get processed faster.

Since your bin is new, make sure you are not overfeeding them.

Thank you for the info Otis11

How long before I start seeing the brown composty looking stuff?

Depends how many worms you have and what size of bin. On the average I would say about a couple of months depending.

I take the liquid and dilute it, to whatever seems suitable to you, by running it through a piece of sheet, to get out the particles that would clog my sprayer, then run some water through the cloth to extract any worm juice and to dilute the juice in the container after the strain. Then I put the solution into a mister and mist the leaves of my plants, foliar feeding them. Good stuff.