Use Alt + Shift + → to expand the selection between braces or tags.
Use Alt + Shift + ← to shrink the selection between braces or tags.
Here is the Microsoft Visual C++ shortcuts cheatsheet that might help you.
Update 2019/3: this inner functionality of Visual Studio is not working very well after some updates. Alt + Shift + → now selects things including braces (which annoys me).
It is not like its behavior before. It selected things between curly braces, when I first posted this answer. I’m using Mark’s answer now.
If anyone has a better solution (without an extension) now, please leave a comment.