> so I infer its meaning. But I don't know if I infer it correctly! I'm convinced this is how we end up with word meaning mutations
Sometimes. Sometimes something more intricate happens.
是 is, in modern Chinese, the verb "to be". Like other Mandarin verbs, it works in pretty much the way an English speaker would expect, coming after the subject of the sentence and before the object: "I am an American": 我 [I] 是 [am] 美国人 [an American].
That's also how verbs work in Old Chinese. It isn't how predication works, though; where modern Mandarin has (thing)是(other thing), Old Chinese says (thing)(other thing)也. "Lao Tzu was a man of Ch'u": 老子 [Lao Tzu] 楚国人 [a man of Ch'u] 也 ["was"]. 是 is an important word in Old Chinese, but it's not even a verb - it's the proximal demonstrative pronoun, "this".
How does that turn into the verb "to be"? Well, "this" is often used to refer back to a complicated noun phrase. So you see a reanalysis:
鱼出遊从容是鱼乐也
鱼 [fish] 出遊 [come out on a pleasure trip] 从容 [(and) relax]。是 [this] 鱼乐 [fish happiness] 也 ["is"]。
鱼 [fish] 出遊 [com(ing) out on a pleasure trip] 从容 [(and) relax(ing)] 是 [is] 鱼乐 [fish happiness]。
The meaning of the utterance didn't change at all (though we forgot that it was supposed to end with 也). But the meaning of 是 inside it changed quite a bit.