There are several key reasons as to its usage, but only one for its role as a mainstay on certain military uniforms found in the world today: tradition.

Historically, however, the "chinstrap" is actually not a chinstrap at all — instead, it is referred to as a "curb-chain". References to the strap as such can be seen evidently in historical works that reference these styles of military outfits, namely in the works of Sir Thomas Henry Hall Caine. In The Christian (1897), Caine writes, "There were soldiers, sentries, policemen, the generals in cocked hats, and the Prince himself in a bearskin, riding by with the jingles of spurs and curb-chain." (44). It's been pointed out that this source is likely referring to the curb-chain of the horse, and not of the bearskin. Here's a source from The Junior Officers' Reading Club: Killing Time and Fighting Wars (2010) by Patrick Hennessey, an author and former British Army officer with firsthand experience, that removes the ambiguity surrounding the name of the piece: "The bearskin, after all, had been a pretty practical bit of kit for dismounted troops holding the line against cavalry — the curb-chain protecting the face from the downward slash of the sabre and the deceptive hollow height of the caps themselves tempting and disarming the lancers." A more historical source from the 19th century, History of the Royal Sappers and Miners (vol. 2) (1857) by Thomas William John Connolly, says the following: "A black bear-skin head-dress, scarlet cloth trousers, and white cloth tunic constitute the uniform. The first is free of embellishment and without feather or plume, but has a gilt curb chain for the chin."

The purposes of chains and chainmail throughout military history is quite well documented in both its use and its utility. If you're interested in reading more about that in particular, I'd recommend the Fighting Techniques series of books — namely Fighting Techniques of the Medieval World (2006) by Matthew Bennett and contributors. In short, chain is exceptional at offering protecting from slashes and virtually impenetrable by blade. Chainmail as was found on armor in combat had been all but phased out by the time the Napoleonic Age rolled around. There are a lot of catalysts for this, but the primary rationale is that chainmail and other forms of metal armor are extremely heavy when worn as a hauberk or on the body for protection. The century prior to the Napoleonic Age also saw a stark departure from medieval-style combat, and so the need for plate and chain armor generally lessened as militaries transitioned to standardized cloth uniforms.

Additionally, the Napoleonic Wars revolutionized the way that wars were fought and won. New strategy and formation was needed to keep up with the changing technology — the battlefield saw new innovations with the likes of muskets and artillery. Military commanders and strategists needed to adapt their techniques around this. Both Blundering to Glory: Napoleon's Military Campaigns (1987) by Owen Connelly and Fighting Techniques of the Napoleonic Age (2008) by Robert B. Bruce and contributors cover this extremely well. The most common thing you'll notice in this era is the presence of line musket infantry. In these line formations, the bearskin headwear was used to give the impression of a larger opponent when infantry was lined up neatly in a row, in an effort to intimidate the enemy. But this fashion statement also meant a trade-off in function. Fluffy bearskin hats couldn't protect the head as well as a helmet could, and safeguarding the face was still seen as a necessity in combat. Bayonets found their place on muskets, and the sabre still remained a common melee weapon throughout the era, so it's no wonder why the curb-chain was adapted as it was. Protecting the face from sword slashes was deemed important enough to warrant the installation of a barrier designed to stop enemies from slashing your face. Chain is light enough to use comfortably in that small form factor, so chainmail experienced a small but slight revival in its place on bearskin hats in the early 19th century and onward. The chain was well-suited for that purpose, and found itself as an alternative to the heavy dragoon helmets or a clunky plate visor that were commonplace up until that point.

Nowadays, the curb-chain is entirely ceremonial. In fact, some of these straps such as those seen here aren't even made out of chain, and offer no protection whatsoever. It's just the way they were worn in traditional military garb, so we continue to wear them that way to this day — even if the function isn't necessarily there anymore!