This is actually pretty easy to do, and one of the first "cool" things an average map maker would get to know.
The mission breifings area is like a box, this is like your canvas, which you can make an animation of. Basically, what you do, is just like making a slideshow, but with really fast intervals. If you know how to make slideshows, think of the mission objectives box as your big white screen for each slide.
The triggeri works like this:
Lets say you wanted to spell "Hello" one letter at a time.
Trigger:
=Set Mission Objectives: H
=Wait for 50 (or whichever interval) milliseconds
=Set Mission Objectives: He
=Wait for 50 (or whichever interval) milliseconds
=Set Mission Objectives: Hel
=Wait for 50 (or whichever interval) milliseconds
=Set Mission Objectives: Hell
=Wait for 50 (or whichever interval) milliseconds
=Set Mission Objectives: Hello
There you have it!
But (not having my SC CD, can't watch your game's breifing), there are many more complicated ways to spell hello, or whatever text you want. So if you want to spell it backwards you could do that. Irregular intervals are cool if used right.
You don't have to limit yourself to one line too, make 4, 5, even 10 lines, and also you can combine this technique with color. The possibilities are endless!

Edit:
I would also like to add, that this technique can also be used in the big center box, where the text is, but that is much more tricky. It is the same, but for how the "Display for XXXX millisecodns", the XXXX is the exact number of how many milliseconds the wait after it will be. Also the center is kind of hard to find.
If you are daring, while using the center box, you can make the text longer than just the initial size of the box is, make a scroll down effect, which is very tricky to use.