It's still a little bit difficult for me to comfortably present a design. It always feels like I'm searching for a structure as the words come out of my mouth. This leads to the presentation coming off scrambled. Is there a method or structure you've found for presenting a new design to clients?
One of my mentors said that you will never die of a bad presentation. So relax, take your time, and practice. :)
You followed a process to create the design. You identified pain points, user goals, issues with the existing design. Walk them through the highlights of that process. I find that summarizing the work gets me relaxed and makes the presentation flow better. Then, the big reveal, where you describe any changes to the process, and describe the design elements.
Hopefully you have had collaboration before this point; they have seen sketches and prototypes; some of the elements were their ideas, even.
My thought is, there should be few surprises at this point. The presentation is a formality for additional stakeholders' benefit.
The key for a good presentation is preparation and rehersal. Ideally you should rehearse aloud. Then define places where you have pauses and other problems. Improve it. You shouldn't learn by route all the text, rather have some key points in mind and convey them. You can write those key points on a palm-size card. They create the skeleton of presentation. So even if your forgot something concerning current point, you could easily switch to next one.
It's worthwhile spending a little time to 'understand the client':
Look at their website: are they 'lots of words' people or 'lots of graphics' ?
(and while you're doing this you can also see what they wear...)
Create an outline, and have your main focus points well-rehearsed.
As an aside, I'm not great at talking in front of groups, and found Toastmasters to be an excellent organization dedicated to improving public speaking and organizing your thoughts. Groups are all over. Chances are there's one in your area.