[git] Could not open a connection to your authentication agent

Read @cupcake's answer for explanations. Here I only try to automate the fix.

If you using Cygwin terminal with BASH, add the following to $HOME/.bashrc file. This only starts ssh-agent once in the first Bash terminal and adds the keys to ssh-agent. (Not sure if this is required on Linux)

###########################
# start ssh-agent for
# ssh authentication with github.com
###########################
SSH_AUTH_SOCK_FILE=/tmp/SSH_AUTH_SOCK.sh
if [ ! -e $SSH_AUTH_SOCK_FILE ]; then
    # need to find SSH_AUTH_SOCK again.
    # restarting is an easy option
    pkill ssh-agent
fi
# check if already running
SSH_AGENT_PID=`pgrep ssh-agent`
if [ "x$SSH_AGENT_PID" == "x" ]; then
#   echo "not running. starting"
    eval $(ssh-agent -s) > /dev/null
    rm -f $SSH_AUTH_SOCK_FILE
    echo "export SSH_AUTH_SOCK=$SSH_AUTH_SOCK" > $SSH_AUTH_SOCK_FILE
    ssh-add $HOME/.ssh/github.com_id_rsa 2>&1 > /dev/null
#else
#   echo "already running"
fi
source $SSH_AUTH_SOCK_FILE

DONT FORGET to add your correct keys in "ssh-add" command.