"If a man among you will bring a korban."
This passuk starts with in the singular-Adam- and ends with a reference to the plural- mikem. The medrash here (4:6) talks about mutual responsibility and group identity of the nation.
A person on a boat takes an chisel and begins gouging a hole in the floor of his cabin. Everyone screams at him to stop, and he answers, “What business is it of yours? I'm making the hole in my own cabin.” The passuk here is telling us that a man’s behavior is “mikem”, he is a part of the organic whole of the nation. His sin affects the community as a whole, by coarsening and polluting the entire world. When he does teshuva and brings a korban, he elevates and benefits the entire briyah.
Same idea— Sanhedrin 27, going on the passuk in Vayikra 26 “vekashlu ish b’achiv,” that “ish b’avon achiv, shekulan areivin zeh bazeh.” This is one of the places that Chazal say "Kol Yisrael areivin zeh Bazeh," usually quoted as "Kol Yisrael areivin zeh lazeh." We are responsible for each other, and we answer for each others' behavior.
One of the practical ramifications of this concept is this: Sometimes, a person can fulfill a mitzva through an agent. Example- blowing the shofar or reading the megilla. However, the agent needs to be obligated to do the mitzva as well. A woman cannot blow the shofar for men, because women are, technically, not obligated to blow the shofar. If so, can a person who has already fulfilled a mitzva do it as an agent for someone else? And can he make a bracha when doing so, because the words of the bracha are "asher kidshanu be'mitzvosav ve'tzivanu," who has obligated us to do this mitzvah," and the agent is no longer obligated. But because of the rule of Arvus/Areivus, the agent is considered to have a current "obligation" to do the mitzvah: the obligation to see to it that others fulfill it. Thus, not only can he be an agent on behalf of the others, he can even say "vetzivanu", because he is still obligated. As long as the others haven't done the mitzva, I haven't fulfilled the mitzva.
The word Areivin and the idea of Arvus has often been associated with the rules of a guarantor or a co-signer on a loan, also called an Areiv.
The sefer Shearim Metzuyanim Behalacha brings from a sefer called Siach Yitzchak the following: The opinion of the Bahag is that if you already fulfilled a mitzvah, you can't be motzi someone if the guy knows how to make a bracha himself. This Sefer explains the reasoning of the Bahag like this: there's no din arvus where the loveh can pay himself. The guarantor of a loan is not liable unless the debtor cannot or refuses to pay. Here too: if the other person is willing and able to make the bracha, you have no arvus at all.
I don't know who this Siach Yitzchak is, but he is wrong for several reasons:
1. The bahag is talking about the bracha, not the mitzvah. If he can't do the mitzva himself, then you have arvus on the mitzvah, and once you're an arev on the mitzvah, you can make the bracha even if the guy could make the bracha himself.
2. See the sefer Mishulchan Gavoah in Nitzavim that brings that Reb Chaim explains the difference between areiv (guarantor) and kablan (co-signer); An areiv is only responsible for results of loveh's default; it is not a liability for the loan, it is a promise to pay that only comes into being when triggered by the debtor's default. A Kablan, on the other hand, is responsible for the loan itself. Since you can use arvus to make a bracha for other person, Reb Chaim says it must be that the arvus of mitzvos is really kablanus. Reb Boruch Ber says this explains the Gemara in Shabbos that when Klal Yisrael is punished for failure to be mekayeim mitzvos, the tzadikim are punished first. Why? As areivim, they should not be first. Answer- they're not just areivim, they're kablanim.
So according to Reb Chaim, the pshat in bahag is wrong, because if we're kablanim, our obligation exists even where the principal is capable of fulfilling his obligation.
On the other hand, it could be that this is gufa the machlokes: Because R' Akiva Eiger there brings from Pri Chadash that we don't hold like the Bahag. What's the machlokes? Maybe the Bahag holds we have a din Areiv, and the Pri Chadash holds we have a din Kablan.