I can point you to a short answer and a couple of further reading options. The short answer is: 'Garibaldimania' seems to have been sweeping England at the time. He was seen as a populist figure, unifying Italy (which only became a country around this time) and defying the pope, and his popularity spread beyond Italian borders to places such as England. Garibaldi visited England in 1864 to huge crowds, as illustrated in this sketch of a gathering in London. Nottingham also felt the pull of this new celebrity: "Nottingham workers met at a now-demolished building on the site of the Council House, the Exchange Hall, where they enthusiastically adopted an address to Garibaldi, controversially praising him for “undoing the chains of oppressed nationalities, and helping to sweep away the wrong, and establish freedom”. (Goldsmith, 2018). Garibaldi wasn't allowed to leave London, but the Nottingham workers managed to send a delegation to him in the south and delivered the address to him.

The Forest kit was adopted the year after Garibaldi's visit, and that Garibaldi red remains in use to this day for the club, but it's important to note it was just one example of Garibaldi's influence throughout England. Left Lion - an alternative print newspaper in Nottingham (linked above under Goldsmith) - notes that there were streets, houses, and pubs named after him, mostly now gone, but Herbert Kilpin, the founder of AC Milan, who was from Nottingham does indeed have a current pub named for him in the center of town. Kilpin would also, in the 1880s, found an amateur football club called the Nottingham Garibaldi.

Other examples of Garibaldimania in England: the Garibaldi currant biscuit, an eponymous sauce in Scotland, all kinds of memorabilia such as china figurines, mugs, plates, and tankards, and the like. Because of the blousy red shirts Garibaldi and his army wore, that style of shirt and color became incredibly popular - Goldsmith notes that three separate places in Nottingham alone supplied these shirts, "all touting the fact these shirts were worn by the local Rifle Volunteers, while one emphasised (sic) their suitability for boating or cricketing." So here - as with most things in history! - context is absolutely crucial, though a lot of it requires historical research rather than wandering the streets of England to see all of the Garibaldi pubs, etc, as would have been in existence in the late 19th century.

So, when the group that founded what would become Nottingham Forest met the following year, they voted to adopt Garibaldi red as their team color, and though the Forest name was in use from the start, they were also called the Garibaldi Reds (a name now used for a Forest fan podcast, too!)

This just scratches the surface - the echoes of Garibaldi's popularity continue through the popularity of football in England (and a trip Forest made to play in Argentina in 1905, which further spread the color and its history into South America), through other teams such as Arsenal (as you say) and AC Milan (via Nottingham-born Kilpin, though, not through a homegrown Italian impulse), and more.

Some resources, beyond those linked:

Derek Beales, 2010. "Garibaldi in England: the politics of Italian enthusiasm," in *Society and Politics in the Age of the Risorgimento (*Cambridge)

Don Wright, 2015. Forever Forest: The Official 150th Anniversary History of the Original Reds (Amberley Publishing)

ETA: Come on you Reds!!