This is happening because of two things:

* `vim` (at least in this case) and `sed`, when doing in place editing, actually delete the original file and then create a new one with the same name. 

* the ability to delete a file depends on the permissions of the directory containing the file, not on the permissions of the file itself. 

So, what is happening here is that you have write permission on the directory, which means you can change the directory's contents, including deleting and creating files. So when you run your `sed -i` or save with `:w!`, you are deleting the original and then creating a new file. This is also why the ownership changes: this is actually a different file. 

You can demonstrate this by checking the file's [inode][1] before and after editing:

```
$ ls -ld foo/
drwxr-xr-x 2 terdon terdon 266240 Nov 16 13:43 foo/
$ cd foo
$ sudo sh -c 'echo foo > file'
$ ls -l 
total 4
-rw-r--r-- 1 root root 4 Nov 16 13:42 file
```

After those commands, I have `file`, owned by root, in the directory `foo/` to which my regular user has write permission. Now, let's use `ls -i` to check the inode, and then make a changed with `sed` and check again:

```
$ ls -l file
26610891 file
$ sed -i 's/foo/bar/' file
$ ls -i file 
26610890 file
```

You can also see `vim` doing the same thing by running 

```
strace vim file 2> strace.out
```
Then editing the file and saving with `:w!`. In your `strace.out`, you will see:

```
 unlink("file") = 0
 open("file", O_WRONLY|O_CREAT|O_TRUNC, 0644) = 4
 write(4, "bar\n", 11) = 11
``` 
So, the file was first deleted (`unlink("file")`), then a new file of the same name was created (`open("file", O_WRONLY|O_CREAT|O_TRUNC, 0644)`) and the modifications I had made were written to it (`write(4, "bar\n", 11)`). 

As you can see above, the inode changed: this is a new file with the same name. So you did not actually change a file you didn't have write access to, you changed a _directory_ to which you did have write access by deleting a file in that directory and then creating a new file in the directory with the same name as the old one.

I have answered a similar question here: https://askubuntu.com/a/815849/85695.

 [1]: https://en.wikipedia.org/wiki/Inode