"Anachnu" means "we."
There doesn't seem to be much more to say about it. We is the plural of I, at least grammatically speaking. And while the Hebrew word does not seem to be directly connected to the singular "ani," it is close enough to make sense. (Actually, you will also find "anu" as the plural of "ani" and "nachnu" as a variant of "anachnu." But there does not seem to be a close relationship between "ani" and "anachnu".)
There is also no defensible connection to other words similar in form to anachnu. "Anacha" means a groan or wail. "La-nu'ach" means to rest. Neither one seems related to anachnu.
But the fact that "we" is a unique word in Hebrew (and in English) is important. After all, though "we" may hold a particular meaning and imply certain repercussions grammatically speaking, it is not really the plural of I in our experience. Every individual is an "I," an entity unto himself or herself. As different as each person is from the next, the pronoun that indicates the individual is common to everyone in the diverse human race.
But just because a bunch of "I's" are in the room doesn't mean they constitute a "we." (The outside observer may consider all the "I's" a "them," but then the "I" who is the observer is not a part of "them.") In order for "we" to exist, there must be something shared among all the others to which every "I" can lay claim. "We" is entirely inclusive. It presumes membership for everyone who is in the group. And in its usage in our prayers, it also presumes that the person or people doing the praying speak for all those who are not praying.
When an "I" becomes part of a "we," he or she belongs to an indivisible family.
If all this seems a little peculiar, it really addresses an essential aspect of Jewish prayer. I have often heard (sometimes from myself) the criticism of Jewish worship that it is fixed liturgy rather than spontaneous and flexible. Where do I fit in, is often the question. Without dismissing that concern, it ignores the membership that every "we" and every suffix and prefix in Hebrew than means our, us and we bestows upon the entire House of Israel, whether each "I" is inspired, expressive, faithful or none of the above.
So I guess there really isn't much more to say about it.