After finishing the game and finding all the postcards, go back to the main menu and click on the notice board in the background that is made up of all the postcard pieces you collected.
Clicking on this will zoom you into the board, which is where you'll be able to see hidden numbers in the backgrounds of some of the postcard images (Tip: You can click on each of the postcards for a larger image display).
After getting these numbers, zoom back out then click on the numbers on the washing machine. This is where you can input those numbers.
Again going back to the postcards on the board, you will see some stick/claw-looking objects in two of the photos (Remember you can click on these images to get a bigger display of them), noting down the direction in which the claw part is facing (answers below if you cannot figure it out), zoom back out and go back to the washing machine. On either side of the number pad are 2 circles with the stick/claw inside of them. Once you've put these in the correct place, press the big yellow button in the middle, then wait a few seconds before pressing the play button again and you'll unlock the secret level!
If you are unable to see the numbers & symbols or want to skip through to this, I've put them below:
*** Spoiler - click to reveal ***You can find the 4 numbers needed in the background of the Sand Pyramid postcard, the Mediterranean Greece postcard, the Medieval Germany postcard and the 90's New York postcard. The numbers are as followed: 6532
As for the stick/claw objects, you can find these on the Wild West postcard and the Future Tokyo postcard. You can see that the Wild West one is facing between the 2 circles, and the Future Tokyo one is facing between the square and circle.
To put it simpler, on the washing machine, the left side needs the claw to be facing directly downwards, and the right side needs the claw to be facing to the left.