One clarification (and a point that confused me):
"remotes/origin/HEAD is the default branch" is not really correct.
remotes/origin/master was the default branch in the remote repository (last time you checked). HEAD is not a branch, it just points to a branch.
Think of HEAD as your working area. When you think of it this way then 'git checkout branchname' makes sense with respect to changing your working area files to be that of a particular branch. You "checkout" branch files into your working area. HEAD for all practical purposes is what is visible to you in your working area.