Get Involved!The typical argument you hear in the US is that most people are lazy, and so won't bother to vote unless forced to. If you make it compulsory, they will get involved in the process and the election will be a better representation of the will of the people.
As an aside, I would argue that the first thing the US should be doing is making elections on the weekend, or even better, make election day a public holiday. People might be lazy, but adding the extra hurdle of having to get away from work to vote certainly doesn't help, and probably disproportionately affects lower income people who have jobs where it is more difficult to get away for a while, even if the employer would like to allow it.
If you make voting compulsory, there is no good reason to expect that most people who are currently disinterested will suddenly care passionately about the electoral process. Some might, but most will probably remain just as disinterested and go and vote with the minimum effort and involvement possible.
Signal to Noise RatioThe biggest problem with compulsory voting is that rather than getting a true representation of the will of the people, you are now muddying the results with a whole heap of votes from people who don't care or are poorly informed. Possibly all these extra votes would be equally distributed across all voting options (if you treat them as random chance) but studies show that this is not the case. For example, the choice at the top of the ballot will be selected more frequently than the choices below.
So adding in a bunch of forced votes will skew the results in some way, giving a less correct result.
Low Information VotersThe other major problem is that of low information voters. This is a problem in any voting system: voters who have poor or limited knowledge of the candidates, issues, and other relevant information for making an informed decision. Typically these are voters who get most or all of their relevant knowledge from sensationalist media, such as biased television news and news opinion shows, lowbrow newspapers, talkback radio, etc. Because these media sources rely on exaggerating, omitting important facts, and often outright lying to get ratings, and because most people will not make the effort to fact check, you end up with horribly misinformed and misguided voters.
Do you really want these people voting? How can they possibly improve the quality of the result?
IndifferenceThen there is the problem of indifference. This can be either people who don't care which candidate is elected, or people who see all options as being equally bad, all candidates as being equally corrupt/untrustworthy. Forcing this person to vote guarantees invalid information since, from the voting perspective, that vote would have the same effect as the vote of a person who is 100% in favour of a particular candidate, yet this is clearly not a fair representation of how they feel about the candidates.
Amplifying Small DifferencesFollowing from the previous point is the general problem that voting turns a fuzzy preference into a black and white one. Many voters are not going to be 100% in favour of one option and 0% in favour of all of the others. In reality they will see pluses and minuses in the different options. At the extreme they will be 50/50 like the indifferent case above, or maybe 51/49, just very slightly preferring one to another.
But the voting process turns all of these votes into a 100% choice for one option. If you instead had people voting on, say, a scale from 1 to 10 for all options, and then normalized so all of their preferences summed to 1, I think the election outcomes may well end up looking very different.
Optimal RepresentationIn the end, I don't think there is any perfect system that will capture the will of the people (and at least in part because I think the 'will of the people' is probably a poorly defined concept). However, it seems to me that the best results will occur when you maximise the number of well informed voters who have a strong opinion on the choices, and minimise the poorly informed voters or the ones that don't feel very stongly about any of the choices. Compulsory voting seems to pretty much get you as far from this ideal as you can go.
If voting is voluntary, but ideally has low barriers (like being held on a weekend as is done here in Australia), then you will get lower turnouts, but the quality of those responses will rise. The purpose of representative government is to have a smaller number of people whose job is to study the important issues and make good choices on behalf of their constituents, who don't have the time to do this. It doesn't seem like much of a stretch to extend this same principle to voting, with low information voters trusting that more informed and passionate people will have a better idea of the best choices, and keeping out of it themselves. Of course we need to look out for people gaming the system, but that's a separate matter.
Tweaking Compulsory VotingIf you must have compulsory voting, then it seems important to me that there is a valid way to vote for people who are mostly indifferent to the available choices, or even more importantly, well informed voters who feel that the available choices are all equally terrible. I think it would be interesting to have some sort of 'no confidence' option on the ballot. If more than a certain percentage of votes are no confidence votes, then you would do something like a re-election, where all current candidates are forced to stand down. There are probably all sorts of issues with a system like this, but I think it would at least force candidates to listen to their constituents and try to be genuinely good choices, rather than the current situation, where they simply try to be the least bad choice.
Actually, this option would probably also be valid for non-compulsory voting systems too. As long as candidates know that one of them must be chosen, the bar stays low and they only have to suck less than the others. But if they know that they can all be thrown out, the game suddenly changes, in our favour.