There's no mystery here. Here's a link to the Snopes page listing both the contents of the email and Snopes' analysis: http://www.snopes.com/politics/obama/money.asp
The short answer is, he took out student loans. Then followed a long, drawn-out process of repaying student loans on professor's and lawyer's salary (not exactly a paltry sum), then huge income from best-selling book to finally pay off those loans.
The only one that's not fully explained so far is the trip in '81, but if you're staying with friends and relatives, even that one is do-able. Nothing nefarious is required.
Imagine that you're a candidate. You have a volunteer organization to raise and collect campaign donations, and these people are hired by yet more volunteers. The plain fact is that you don't even know the people who are working on your behalf. You accept funds on the Internet, by phone, by check.
How are you, personally, supposed to know that any of these donors are who they say they are? The best you can really do is return money if irregularities crop up.
I think it highly unlikely that, even if he knew all these illegal donors personally, Obama would have solicited campaign donations from them. First of all, at this point he doesn't need the money: he has enough donations from legitimate Democrat donors. Second, this sort of connection harms his campaign more than the money helps. If he were a real evil genius he'd have told these people to shut up and lie low. The "fault" of these donations lies with the donors, not the candidate. If you make it the candidate's fault, then you can't defend McCain when similar things happen without his knowledge.
What IS disturbing Obama's relationship with William Ayers, known unrepentant terrorist. Obama can't convincingly pretend that he didn't know about this guy. Obama was a professor. He was a lawyer; well educated, knowledgable and highly interested in political affairs. And yet he's perfectly willing (despite all his claims to exerience) to imply that he was naive and stupid enough to sit in Ayers' kitchen and plan his political career. So depending on whether you choose to believe Obama's story, either his judgment is completely suspect, he's a conspirator, or he actually IS naive and ignorant. Any of these conclusions validates the Republicans' opinion of him. None of these conclusions recommends him for President.
UPDATE (2008-10-09): As it turns out, the link between Obama and Ayers is much stronger than initially thought. Read about it in American Thinker. Better yet, read about it in the Wall Street Journal. Obama's campaign wants to distance themselves from Ayers, stating that "any attempt to connect Obama with events of almost 40 years ago is ridiculous." The problem for Obama is, nobody's trying to do that. They're examining the public record and connecting him with events of which he himself was an active participant. And these events are troubling: personally, I would characterize it as no less than an attempt to subvert the educational system and indoctrinate children. As Stanley Kurtz rightly observes, that is a story "even if Ayers had never planted a single bomb 40 years ago."What's also disturbing is the quality of supporter that Obama attracts. It's understandable that people in the Middle East support him. The guy's name is Arabic, after all, and Libya's Qadafi thinks he's a "good Muslim" on that basis alone. But he's gathering support from terrorists, hostile governments, and a wide lunatic fringe. There's a reason for that. I think it's pretty clear that they know he's naive; they can see with their own eyes from the Ayers connection that he's sympathetic to terrorists (and there's no denying that he's been sympathetic to at least one); and they think he's a pushover. Hostile governments have no reason whatsoever to support a US candidate unless they are convinced in their minds and hearts he is bad for this country. I happen to agree with them... a Barack Obama presidency would be a disaster, and a gift to the Taliban and Al Qaeda.
But these are largely questions about other peoples' support of Obama, which he can't fully control. Beyond these connections, what disturbs me the most is the things that he can directly control. The fact is that for all his talk of "change", Obama has few if any votes or actions that support his talk. In the debate and in all his stump speeches he looks back for blame for the economic situation and Iraq (consistently mis-assigning blame). For months he's been running against George Bush, NOT John McCain. He picked a left-wing career Capitol insider as his running mate. He hasn't even made it to the PRESENT, much less looked to the future. His campaign slogans are quite simply untrue. They're not supported at all by his actions.
By contrast McCain looks to solutions. He has been pushing his ideas for years, and only now have the times caught up with him. He chose a running mate with an eye to the future, and he picked her from far outside the Capitol Beltway. He walks his talk.