This answer may be a little superficial, but I'm not quite how much detail you are asking for here.
"Sprinklers" were just bottles, usually with long-narrow necks, which we believe to have been used to sprinkle scented water, perfumes or or other liquids. Depending on the burial conditions, it is sometimes - but by no means always - possible to carry out residue analysis to determine what they originally contained.
By analogy with the later attested use of similar vessels, they were probably used to sprinkle their contents over a relatively large area.
The design of the bottle varied considerably over time and in different places. We have found examples over a geographic range stretching from the UK to India.
Bottles with long, narrow necks that are often called "sprinkler bottles" have been around since the Roman period. Their use can certainly be attested from the 13th century. This page from Behind the Scenes at the Getty includes an example of a late 14th / early 15th century sprinkler flask from Byzantium which:
"would once have been used to sprinkle myrrh and holy water."
A nice example of a 13th century Islamic sprinkler bottle was discovered a few years ago during the archaeological excavations ahead of the construction of the Leadenhall building in London. I'm not sure if any residue analysis has been carried out on this example, but according to the article, it would have:
"contained perfume or rose-water"
and can be seen in this National Geographic article.