Change the shell to a different bash version at /usr/local/bin/bash

Bash only reads scripts by default in your home directory, or if they are missing in /etc. See Bash's documentation. If using OSX's Terminal.app then by default it reads ~/.bash_profile.

This is true of all bash on Linux or other OSes.

To read from another directory e.g. /opt/local/etc/bash_completion.d you have to edit your start up files to source (i.e. include) the files from there

The instructions for the script do not mention /opt/local/... which is a non standard location on any Unix. (by non-standard it is allowed to be used by third party packages but not defined what should be in there) They say

1. Copy this file to somewhere (e.g. ~/.git-completion.sh).

2. Add the following line to your .bashrc/.zshrc:

   source ~/.git-completion.sh
   

3. Consider changing your PS1 to also show the current branch, see git-prompt.sh for details.

The progit quote assumes you know bash. All it is saying is put the files from the first quote into a specific place if you want all users on the machine to use them and not in ~ where only the user installing it can see them. i.e. the point is multi user versus single user.

He also picks that path as nothing else uses it and you should not edit /etc files in OS X as Apple's OS upgrades could overwrite them so you need to choose another place. (I would have chosen something under /usr/local as there is where manually maintained scripts are meant to go).

~/.bashrc is the correct place to edit and add the source. See your other question and the bash manual for setting up ~/.bash_profile The bash suggested way is source .bashrc into .bash_profile. Note that on OS X using Terminal.app is not the one way of running shells so there can be sessions starting with .bashrc.

Your prompt is messed up, specifically PS1:

PS1=$[\033]0;37m]

that's missing a lot of escape characters (\e[) needed for the colors (and most useful parameters for a PS1). That's also why you get the 37m in the terminal window title. Try setting it to something different by running:

export PS1="\e[0;31m[\h:\W \u]\$\e[m "

and see if that works. It should give a red (thats \e[0;31m) prompt showing hostname (\h), current working directory (\W) and logged in user (\u) inside brackets [] and the bash exit status of the previous command (\$). Note that at the end the color is reset to the default of the session with \e[m .

If the above worked, you only have to find out from which configuration file your "bad" PS1 comes from: look for an "export PS1=" line in ~/.profile, ~/.bash_profile, ~/.bashrc, (as M K already suggested in his answer) and put in the above version.

There are a lot of answers around here with helpful colors codes and inputs for configuring the PS1, like this one for example.