Error 1013: The private attribute may be used only on class property definitions

You seem to have some code misplaced.  Just after your Main function you have a line of code...

addEventListener(Event.ENTER_FRAME,onEnterFr);

You probably want to move that inside one fo the functions.

As far as the line you say is the problem, you appear to have that function nested within the placeSnake function.  Do not nest functions like that.

does this solve your problem

package

{

          import flash.display.Sprite;

          import flash.events.Event;

          public class Main extends Sprite

          {

                    private const FIELD_WIDTH:uint = 16;

                    private const FIELD_HEIGHT:uint = 12;

                    private const TILE_SIZE:uint = 40;

                    private var the_snake:the_snake_mc;

                    private var snakeDirection:uint;

                    private var snakeContainer:Sprite = new Sprite();

                    private var bg:bg_mc = new bg_mc();

                    public function Main()

                    {

                              addChild(bg);

                              placeSnake();

                              addEventListener(Event.ENTER_FRAME, onEnterFr);

                    }

                    private function placeSnake():void

                    {

                              addChild(snakeContainer);

                              var col:uint = Math.floor(Math.random() * (FIELD_WIDTH - 10)) + 5;

                              var row:uint = Math.floor(Math.random() * (FIELD_HEIGHT - 10)) + 5;

                              snakeDirection = Math.floor(Math.random() * 4);

                              the_snake = new the_snake_mc(col * TILE_SIZE, row * TILE_SIZE, snakeDirection + 1);

                              snakeContainer.addChild(the_snake);

                              switch (snakeDirection)

                              {

                                        case 0: // facing left

                                                  trace("left");

                                                  the_snake = new the_snake_mc((col + 1) * TILE_SIZE, row * TILE_SIZE, 6);

                                                  snakeContainer.addChild(the_snake);

                                                  the_snake = new the_snake_mc((col + 2) * TILE_SIZE, row * TILE_SIZE, 6);

                                                  snakeContainer.addChild(the_snake);

                                                  break;

                                        case 1: // facing up

                                                  trace("up");

                                                  the_snake = new the_snake_mc(col * TILE_SIZE, (row + 1) * TILE_SIZE, 5);

                                                  snakeContainer.addChild(the_snake);

                                                  the_snake = new the_snake_mc(col * TILE_SIZE, (row + 2) * TILE_SIZE, 5);

                                                  snakeContainer.addChild(the_snake);

                                                  break;

                                        case 2: // facing down

                                                  trace("down");

                                                  the_snake = new the_snake_mc((col - 1) * TILE_SIZE.row * TILE_SIZE, 6);

                                                  snakeContainer.addChild(the_snake);

                                                  the_snake = new the_snake_mc((col - 2) * TILE_SIZE.row * TILE_SIZE, 6);

                                                  snakeContainer.addChild(the_snake);

                                                  break;

                                        case 3: // facing right

                                                  trace("right");

                                                  the_snake = new the_snake_mc(col * TILE_SIZE, (row - 1) * TILE_SIZE, 5);

                                                  snakeContainer.addChild(the_snake);

                                                  the_snake = new the_snake_mc(col * TILE_SIZE, (row - 2) * TILE_SIZE, 5);

                                                  snakeContainer.addChild(the_snake);

                                                  break;

                              }

                    }

                    private function onEnterFr(e:Event)

                    {

                              var the_head:the_snake_mc = snakeContainer.addChildAt(0) as the_snake_mc;

                              var new_piece:the_snake_mc = new the_snake_mc(the_head.x, the_head.y, 1);

                              snakeContainer.addChildAt(new_piece, 1);

                              var the_body:the_snake_mc = snakeContainer.getChildAt(2) as the_snake_mc;

                              var p:uint = snakeContainer.numChildren;

                              var the_tail:the_snake_mc = snakeContainer.getChildAt(p - 1) as the_snake_mc;

                              var the_new_tail:the_snake_mc = snakeContainer.getChildAt(p - 2) as the_snake_mc;

                              the_head.moveHead(snakeDirection, TILE_SIZE);

                              // brute force

                              if (is_up(new_piece, the_head) && is_down(new_piece, the_body))

                              {

                                        new_piece.gotoAndStop(5);

                              }

                              if (is_down(new_piece, the_head) && is_up(new_piece, the_body))

                              {

                                        new_piece.gotoAndStop(5);

                              }

                              if (is_left(new_piece, the_head) && is_right(new_piece, the_body))

                              {

                                        new_piece.gotoAndStop(6);

                              }

                              if (is_right(new_piece, the_head) && is_left(new_piece, the_body))

                              {

                                        new_piece.gotoAndStop(6);

                              }

                              // end of brute force

                              snakeContainer.removeChild(the_tail);

                    }

          }

}

Nesting a function is when you place one function within another function as you show in your posting.  What _spoboyle has offered is the unnested version of it.

Errors are like that... some errors take precedence such that othersa are not reported at first.  It looks like you have numerous errors that you need to fix.  You will need to tackle them one at a time, so consider it an opportunity to learn.