Bash script with “execute” command and strange regexp

So I was looking through Paul Irish's dotfiles and found reference to this bash script:

#!/bin/bash

cd "$(dirname "$BASH_SOURCE")" \
    && source 'utils.sh'

declare -a FILES_TO_SYMLINK=(

    'shell/bash_aliases'
    'shell/bash_exports'
    'shell/bash_functions'
    'shell/bash_logout'
    'shell/bash_options'
    'shell/bash_profile'
    'shell/bash_prompt'
    'shell/bashrc'
    'shell/curlrc'
    'shell/inputrc'
    'shell/screenrc'
    'shell/tmux.conf'

    'git/gitattributes'
    'git/gitconfig'
    'git/gitignore'

    'vim/vim'
    'vim/vimrc'
    'vim/gvimrc'

)

# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

main() {

    local i=''
    local sourceFile=''
    local targetFile=''

    for i in ${FILES_TO_SYMLINK[@]}; do

        sourceFile="$(cd .. && pwd)/$i"
        targetFile="$HOME/.$(printf "%s" "$i" | sed "s/.*\/\(.*\)/\1/g")"

        if [ -e "$targetFile" ]; then
            if [ "$(readlink "$targetFile")" != "$sourceFile" ]; then

                ask_for_confirmation "'$targetFile' already exists, do you want to overwrite it?"
                if answer_is_yes; then
                    rm -rf "$targetFile"
                    execute "ln -fs $sourceFile $targetFile" "$targetFile → $sourceFile"
                else
                    print_error "$targetFile → $sourceFile"
                fi

            else
                print_success "$targetFile → $sourceFile"
            fi
        else
            execute "ln -fs $sourceFile $targetFile" "$targetFile → $sourceFile"
        fi

    done

}

main

There are two things in this script that really confuses me.

Firstly, what does this sed actually do?

sed "s/.*\/\(.*\)/\1/g"

Secondly what does execute do?

Couldn't find anything on the execute command.