Easing functions allow you to apply custom mathematical formulas to your animations. For example, you may want an object to realistically bounce or behave as though it were on a spring. You could use Key-Frame or even From/To/By animations to approximate these effects but it would take a significant amount of work and the animation would be less accurate than using a mathematical formula.
Currently, there are 11 new easing functions that you can apply to your animations. You can apply these using Blend 3 beta or you can use XAML or procedural code.
To show you the code way of working with these new functions, here's a relatively crude sample that introduces each of the new easing functions and allows you to change their properties: Easing Function Sample.
Also, I added a section on easing functions in the Animation Overview with some code examples and further explanation.