i adjectives - I know that the word 広し means 広い, but which rule or grammar is included?

The sentence is from my textbook　みんなの日本語

世界広しといえども ...

that means

世界が広いと言っても ...

right?

But what function has it when you use 広し instead of 広い?

During some research I found this Website/ online 辞書 but my Japanese is not good enough to understand it.

Answer

The basic grammar

The し ending on adjectives is the Classical Japanese 終止形【しゅうしけい】 or "terminal form", i.e. the conjugation to use when the word comes at the end of a sentence. The Classical Japanese 連体形【れんたいけい】 or "attributive form" (the conjugation to use when modifying a noun or other substantive) is き. For modern adjective 広い, the 終止形 and 連体形 are the same thing -- both end in い. For Classical Japanese, the 終止形 is 広し and the 連体形 is 広き.

Classical Japanese forms still show up, especially when a writer (or more rarely, a speaker) is trying to impart an archaic or more-formal feel. This might be vaguely analogous to an English writer or speaker avoiding contractions and restructuring sentences to avoid prepositions on the end (such as replacing "who'd you give it to?" with "to whom did you give it?"). The Classical Japanese forms are even more archaic and formal than this English example, but still understandable by modern speakers and readers.

The grammatical construction in your textbook

Looking at your specific textbook example, it's worth noting that this construction of [ADJ ROOT] + し + といえども is a kind of somewhat-formal set construction pattern. いえども itself is Classical Japanese for いっても, as you correctly guess, and this Classical set-construction usage then influences the conjugation of the adjective to also use the Classical conjugation. This is not uncommon in written Japanese, but you generally won't hear this in spoken Japanese, outside of rare contexts like speeches. In spoken Japanese, the [ADJ ROOT] + い + といっても construction is much more common.