The precise process from manuscript to distribution may be different from publisher to publisher - in fact, explaining what to expect is probably a major purpose of such a meeting.
In many ways, this meeting will be similar to a job interview - and the advice will be the same. Don't stress out too much - you're already on their good side (or they wouldn't be arranging all this), and they know you're new to the industry; you'll be nervous and you'll have concerns and you'll want stuff explained, and that's absolutely fine.
They'll probably want to explain the process; this is a great time to ask any questions that you have about what to expect. Come ready with questions if you can. It's good if you can read up on children's publishing a bit (if you haven't already), so things won't be completely new to you, and so you'll notice if anything seems unusual or off-kilter.
They'll probably want to get a personal impression of you; this needs some social skill, but nothing extravagent. They'll want to see that they can have a pleasant conversation with you, that you seem on top of your work and of their expectations of you, that you don't have any major issues which would make working with you a problem. I'll be honest - more than anything, they'll be looking for warning signs, like "this author refuses to accept any criticism," "this author swears like a sailor," or "this author appears mentally disturbed." This kind of thing is a deal-breaker. Of course, anything you can impress them with - professionalism, wit, dazzling beauty, what-have-you - is a very nice bonus.
They might already talk about the contract. Don't commit to anything you're not absolutely certain of - contracts are serious stuff. Handling the contract itself is a whole seperate issue, which you should come into prepared. I won't expand on this here, but be sure you expand on it elsewhere...
For extra reading, here's one of many available essays describing what makes a dream author. A lot of these will only be self-evident when actually working together, but this might give you a certain ideal to aim for.