There are a few things you need in order for a load balancer to work right. It sounds like the thing you're mostly interested in is how to get the right code onto each server. Fortunately this is the easy part!
Use a web-accessible version control service.
There are lots of hosted options, and you can even host your own if you like.
The choice of version-control system is maybe less open-ended; git is strongly recommended although SVN is ok and slightly faster to learn.
Using hosted version control doesn't mean that your code has to be public, although if you want your code to be private you'll need to figure out how your servers will read it. More on that later.
Finally, tell your servers to download the code from the repository. You can do this once, with the understanding that every time you make changes you'll need to ssh into the boxes again to tell them to update, or you can set up a cron job to keep them updated all the time.
Other stuff:
Your load balancer needs to detect if one of the servers goes offline.
It sounds like you have a managed load balancer solution on hand, so it probably has tools for this purpose built in.
You need a shared persistent storage of some kind.
The first, and for some systems only, piece of this is to have your DB on a separate server. (Making the DB redundant too is its own topic.) Depending on your needs, that's often sufficient.
You man need a system for handling secret configuration data.
For example, it's not a great idea to store the DB credentials (or version-control credentials) in version-control. You can keep them up-to-date manually, or you can look into something more complicated.
I've glossed over a lot of best-practice details, which matter, both to keep this short and because what's right for you depends on your situation. I've also neglected a lot of very nice commercial options for solving these problems, both because I'm trying to suggest small changes from your current setup and because capitalism is a disease.
