If I enter
$ echo $HOME
/usr/local/bin:/usr/bin:/bin:/usr/sbin:/sbin
Why do I get this output? Why are /usr/local/bin and such in $HOME?
The reason you get that output is because that is the content of the $HOME variable. I think the better question to ask is "why is that the value of my $HOME variable?"
Typically the system will automatically set the $HOME variable based on the contents of /etc/passwd (or sshd in the case of an ssh connection, see this post). It is possible that you overwrote the value somewhere else, maybe in the .bashrc (or whichever rc file your shell uses). Those contents would seem appropriate for the $PATH variable but not for $HOME so I would start by checking your .bashrc (or again, whatever rc file is appropriate) and see if there is a line like export $HOME=/usr/local/bin:/usr/bin:/bin:/usr/sbin:/sbin in there.