What does ${PATH:+:${PATH}} mean?

The :+ is a form of parameter expansion:

${parameter:+[word]} : Use Alternative Value.

If parameter is unset or null, null shall be substituted; otherwise, the expansion of word (or an empty string if word is omitted) shall be substituted.

In other words, if the variable $var is defined, echo ${var:+foo} will print foo and, if it is not, it will print the empty string.

The second : is nothing special. It is the character used as a separator in the list of directories in $PATH. So, PATH="/usr/local/bin:/usr/bin${PATH:+:${PATH}}" is a shorthand way of writing:

if [ -z "$PATH" ]; then
    PATH=/usr/local/bin:/usr/bin
else
    PATH=/usr/local/bin:/usr/bin:$PATH
fi

It's just a clever trick to avoid adding an extra : when $PATH is not set. For example:

$ PATH="/usr/bin"
$ PATH="/new/dir:$PATH" ## Add a directory
$ echo "$PATH"
/new/dir:/usr/bin

But if PATH is unset:

$ unset PATH
$ PATH="/new/dir:$PATH"
$ echo "$PATH"
/new/dir:

A : by itself adds the current directory to the $PATH. Using PATH="/new/dir${PATH:+:$PATH}" avoids this. So sure, you can use PATH="${PATH:+${PATH}:}/usr/local/bin:/usr/bin" if you want to, or you can use PATH="$PATH:/usr/local/bin:/usr/bin" if you prefer. The only difference is that the former might add an extra :, thereby adding your current directory to your $PATH.

You're correct, it does mean 'if $PATH exists — and is not null — then add :$PATH'.

You need to check whether $PATH exists because you don't want to add the leading (or trailing) colon if $PATH is undefined. A zero-length (null) directory name in the path, as in :/usr/local/bin:/usr/bin, or /usr/local/bin:/usr/bin:, or /usr/local/bin::/usr/bin, means search the current directory.

Excerpted from man bash:

   PATH   ...
          A zero-length (null) directory name in the value of PATH indicates 
          the current directory.  A  null  directory name may appear as two 
          adjacent colons, or as an initial or trailing colon.
          ...

That's probably not what you want to do.

The following two lines do the same thing:

PATH=":/bin"        # search current directory, then /bin
PATH=".:/bin"