Where in this Brother printer does this part go?

According to the Setup Guide:

It appears to be a holder for the print head to keep it from being jostled while in transport. It's removed when the ink cartridge is installed.

If you transport or ship the printer, the ink cartridge should be removed and this part temporarily reinstalled to protect the head in transit.

I could be wrong, but on every new printer I've unpackaged (which is about 4, but still) there have been orange-colored pieces just like that one protecting various delicate parts of the printer. That part probably exactly fits the ink cartridge intake adapters (please confirm), and the rest of the part looks like a handle to pull on (i.e. to remove the piece from the printer). It looks just like it is one of those disposable protection parts intended to be thrown away when you are first unboxing the printer.

An answer to this question from a more general perspective:

• note that these are 3+1 cylinders/nozzles - the fourth one is at a different spacing. Now, what does this remind you of? That's right, CMYK Cyan-Magenta-Yellow for color, and Black.
• Your printer is an Ink Jet printer, so you would expect ink containers for these four colors. This part would likely go on top of them for protection - which makes, sense because they're sensitive and delicate parts.
• Now, where would that ink cartridges be? When they work, they go across the page, back and forth. So probably on one of the sides. Also, they would probably be covered by some piece of stirdy plastic, so as not to to be hit by anything accidentally. This cover must be retractable, or removable, as otherwise you wouldn't be able to access it (although theoretically there could be some button which moves the cartridges out to the middle for manipulation.

And there you have it. The speculation fits what @JustinPearce describes.

_{PS - The object containing the ink cartridges which moves across the page is the "printing head".}

I sell brother printers. That's a part to be thrown away when setting up the printer. It's supposed to keep the internal plumbing sealed. There is ink in the printer plumbing from the onset and that part keeps the ink from drying up until you install the ink cartridges.