One of the things that probably 99% of illustrators (in any field) will have to do at some point is to get out there in the world and show off what they can do. When I started out, it was working for local companies mainly and I gradually expanded around the country and then, with the advent of the web, around the world. Whether you're showing a client your work in your folder, or doing it 'virtually' on a website gallery, the principles are the same. You've got to show off your work in the best light possible.
The first thing I'd say would be to really think about what kind of work you're looking for, and what kind of work you're actually good at. What is it that you really want to illustrate: Books? Children's books? Leaflets and instructions? Diagrams? Think about the work you've done to date and the kinds of work you specialize in. I found it was much easier to target companies who I knew would be interested in the kind of work I did. Going for each and every kind of illustration is hard work and doesn't work very well.
*TIP* Remember: It pays to specialize! This is your career - what is it you really want to be drawing?
It's likely that you've got a pretty VAST collection of artwork that you want to show your clients. If you're just getting started it will probably be mostly student work. There's no problem with that. I found that I could get work from my student work alone, even though I didn't have much experience. Now comes the hard part, picking out the very best bits.
The most important thing is to be selective. While whoever you're showing your work to will have set aside time for you, they're also likely to be very busy. One mistake I used to make was to try to show everything I'd ever done when I went to meetings. If one project generated maybe 20 images, show the best 2. If you're still too close to the work, and can't pick out the best ones, get someone to help you.
*TIP* Even family members can often help pick out blazing examples of good work.
Try to pick pieces which are unique to you. Whether it's a small to medium-sized company or a larger publisher, they'll want to have an idea of what your style is so that they can match you up to relevant jobs. Your portfolio should show the best examples of your work and show them what your style is, and what you are best as. It's likely they'll already have a number of illustrators so it will help if you can offer something unique. That's where style and individuality comes in - which I'm sure you've got lots of!
How you present your work in your portfolio to clients is, in my humble little opinion, far less important than was drilled into our heads in school. So long as your work is in a nice clean and tidy folder that will be fine. I've shown clients loose-leaf drawings before from a cardboard folder. What matters is the work, so don't worry that every drawing you have isn't framed in expensive cardboard and beautifully mounted.
Before you go for your first meeting, or any meeting for that matter, have a trial run. Get your folder up on the kitchen table and make sure that everything is there, and that the pages turn, and that you know where everything is. There's nothing worse than going to a meeting and opening your folder only to have your precious work spill out all over the place. Be organized and business-like, and secure everything in the folder.
On the day be confident, and believe in your work. Remember - illustrators have the luxury of having a portfolio to do the hard work in the interview for them. You're just there to turn the pages. ;-)