Normally, the content of files in the annex is prevented from being modified. (Unless your repository is using direct mode.)

That's a good thing, because it might be the only copy, you wouldn't want to lose it in a fumblefingered mistake.

# echo oops > my_cool_big_file
bash: my_cool_big_file: Permission denied

In order to modify a file, it should first be unlocked.

# git annex unlock my_cool_big_file
unlock my_cool_big_file (copying...) ok

That replaces the symlink that normally points at its content with a copy of the content. You can then modify the file like any regular file. Because it is a regular file.

(If you decide you don't need to modify the file after all, or want to discard modifications, just use git annex lock.)

When you git commit it will notice that you are committing an unlocked file, add its new content to the annex, and a pointer to that content is what gets committed to git.

# echo "now smaller, but even cooler" > my_cool_big_file
# git commit my_cool_big_file -m "changed an annexed file"
add my_cool_big_file ok
[master 64cda67] changed an annexed file
 1 files changed, 1 insertions(+), 1 deletions(-)

For more details on working with unlocked files vs the regular locked files, see unlocked files.