The cursor position is saved (only in Open Document Files i.e. files ending with .ODT) if you fill in the personal information(first and last name are enough). Go to Tools, Options, User Data.
When the user is identified, LibreOffice will open the document where the user left off. This makes sense because a different user opening the same document will probably want to start from the beginning. If there is no user identification LibreOffice assumes it is a new user and always opens from the beginning.