How to make expandable text on clicking more button in views?
This will require some custom theming to accomplish:


In your view, don't trim the field, just show it all for now. This will mean if a user has JavaScript disabled they will still have access the all the content.
In order to enhance the interface as you've described for users with JavaScript enabled, you should take a look at this tutorial. 
You will have to tweak things to get it working with your theme. If you're unsure about adding Javascript to a theme, check this out.