What does sudo -H do?

Generally

man sudo (the exact text may vary, but it will be similar):

-H

The -H (HOME) option requests that the security policy set the HOME environment variable to the home directory of the target user (root by default) as specified by the password database. Depending on the policy, this may be the default behavior.

So why is this even an option? Normally using "sudo" does not change the $HOME environment variable.

for example:

 echo $HOME $USER
/home/testuser testuser

 sudo bash -c 'echo $HOME $USER'
/home/testuser root

 sudo -H bash -c 'echo $HOME $USER'
/home/root root

You can see that a normal sudo changes which user I am from "testuser" to "root", but not what $HOME is set to, while a sudo -H also changes the variable from "my" home directory to root's home directory.

In your Case

pip is warning you that it was executed as the user root and wanted to modify things in $HOME, which was set to '/Users/petertao', which is not owned by root (most likely the "petertao" user). the warning indicates that pip uses $HOME to cache files, but has disabled its own caching because of the folder ownership discrepancy.

Of course while executing as root pip can modify '/Users/petertao/Library/Caches/pip' because root is (almost) almighty. This can become troublesome later because a program running without root could no longer overwrite or modify these files. Instead pip refuses to write to a directory owned by another user.