I don't stop people from answering, but you might find a answer here http://www.reddit.com/r/AskHistorians/comments/284eqa/can_someone_explain_to_me_the_reason_the_us_has/
Though the linked thread includes an anecdote about how America's litigious nature traces to the pre-founding era, I think it's accurate, and fair, to note how much has changed recently. I'll try to avoid bumping up against the twenty-year rule but some might be inevitable, as several recent developments are the result of earlier trends.
Note that this discussion is about the increasingly litigious nature of criminal and corporate legal practice. I won't get in to the McDonalds case/tort law speculation, except to say that that case shouldn't be dismissed out of hand as an example of "what's wrong" with American law. There're plenty of sources on that issue that you could consider.
Statute over Common Law
One trend relevant to your question is that federal "substantive" law -- that is, federal rights arising under acts of Congress, etc. -- have multiplied enormously since the 1930s. Arthur Miller, who helped draft the modern Federal Rules of Civil Procedure -- and interestingly enough was the inspiration for the antagonist in The Paper Chase, the famous book about law school -- chronicles this and other factors leading to the proliferation of federal case in law in several of his law review articles, one of which is linked here. Examples of significant federal legislation include (but are not limited to) the Securities and Exchange Acts, and 42 U.S. Code ยง 1983, the federal statute that creates a remedy* for infringement of federal rights under color of law. That section was enacted in its modern form in 1979. As a prosecutor, it's one we see a lot, and responsible for quite a bit of prisoner litigation.
A similar trend is described by the Federal Rules of Civil Procedure, which were enacted in the 1930s and have been expanded, clarified, etc., ever since. They were created to simplify the rules of pleading and allow easier access to the courts by common citizens. The flip side of this, of course, is that access to the courts is easier for common citizens!
Furthermore, the Rules reduced to statute what was previously handled by the common law: the basic rules of pleading and civil actions. By codifying these matters, the Rules (arguably) made civil procedure more about the letter of the law and less about ancient common law notions of equity and fair play, creating a jurisprudence that is more technical and voluminous. A history of the development of doctrines and how they've changed under the FRCP can be found in any significant Supreme Court decision on the matter; Bell Atlantic Corp. v. Twombly , 550 U.S. 544 (2007) and Ashcroft v. Iqbal, 556 U.S. 662 (2009), are representative examples.
All of these combine to make federal litigation easier to get in to, and more common, but more complicated to do right. In other words, they make law more "litigious" :).
Criminal Law
I'll speak only briefly on this because some of it is too recent for consideration here. Criminal law is highly litigious. Congress has acted time and again to limit the federal rights of state prisoners, believing this to be a way to solve the considerable strain placed on federal courts by post-conviction applications from state defendants, with some success, depending on how you define the term.
The most recent occasion was the Antiterrorism and Effective Death Penalty Act ("AEDPA"), passed by Congress in 1996. Basically, the heart of AEDPA, 28 USC 2254, makes it increasingly difficult for state law prisoners to get a second bite at the apple when raising a federal constitutional claim. AEDPA means that state law prisoners lose more often and at an earlier phase when they raise federal constitutional claims. It DOES NOT, however, mean they can't file such claims. In other words, it makes claims easier to resolve but has not stopped them from proliferating. Query, then, whether AEDPA has accomplished its stated purpose of making the criminal law less litigious and burdened by frivolous applications.
AEDPA also means deportation is easier for aliens convicted of a crime. That's had a dramatic effect on deportations, increasing them massively since reforms began in 1994, and especially since 1996. A recent New York Court of Appeals case discussed this in depth. This means more litigation in some fora -- immigration court and state courts, because it discourages plea bargaining -- but less litigation, and easier litigation, when the government wants someone deported.
(Source: personal practice, and Hart & Wechsler: the Federal Courts & the Federal System)
Discovery
The other matter working to complicate civil practice is "discovery," meaning, how litigants get documents from each other relevant to their dispute. If you sue someone for botching repair of your car, that won't be too involved. All you need to determine the matter might be pictures of your car, and a single contract, and the testimony of a few witnesses. But if you sue Major American Bank, say, for mortgage fraud, how will you prove it? The answer is that there must be a system under which you can gain the documentation necessary to prove precisely how you've been wronged. This is "discovery," or, simply put, document-swapping between litigants.
The federal rules, discussed above, made discovery easier to do.. But they also made it more complicated, especially between sophisticated parties, because there's just more ground to argue over now. It's also difficult to overstate the impact technology has had on discovery. What, from your hard drive, am I entitled to as I try to prove my mortgage fraud case? How will I get access to it? Who will review it and under what rules? How will it be introduced in Court, if at all?
All of these are fertile areas of litigation, and make the transaction costs for business litigation all the harder to bear. For a brief summary of how and why, consider reading the Zubulake series of cases. (Apologies for the Wikipedia. It's just the most convenient source as the "series" involves multiple cases.)