I chose these websites just because they were the first I found however there are probably close to a hundred thousand other law firms in America that you could put in its place. There were 19000 results for New York alone.
In "Letters from an American Farmer" De Crèvecœur wrote about how litigious the American people are:
"As freemen they will be litigious; pride and obstinacy are often the cause of law suits; the nature of our laws and governments may be another. As citizens it is easy to imagine, that they will carefully read the newspapers, enter into every political disquisition, freely blame or censure governors and others. As farmers they will be careful and anxious to get as much as they can, because what they get is their own. As northern men they will love the chearful cup. As Christians, religion curbs them not in their opinions; the general indulgence leaves every one to think for themselves in spiritual matters; the laws inspect our actions, our thoughts are left to God. Industry, good living, selfishness, litigiousness, country politics, the pride of freemen, religious indifference, are their characteristics. If you recede still farther from the sea, you will come into more modern settlements; they exhibit the same strong lineaments, in a ruder appearance. Religion seems to have still less influence, and their manners are less improved." (Crèvecœur, 1997, 46)
"Now we arrive near the great woods, near the last inhabited districts; there men seem to be placed still farther beyond the reach of government, which in some measure leaves them to themselves. How can it pervade every corner; as they were driven there by misfortunes, necessity of beginnings, desire of acquiring large tracks of land, idleness, frequent want of economy, ancient debts; the re-union of such people does not afford a very pleasing spectacle. When discord, want of unity and friendship; when either drunkenness or idleness prevail in such remote districts; contention, inactivity, and wretchedness must ensue. There are not the same remedies to these evils as in a long established community. The few magistrates they have, are in general little better than the rest; they are often in a perfect state of war; that of man against man, sometimes decided by blows, sometimes by means of the law; that of man against every wild inhabitant of these venerable woods, of which they are come to dispossess them." (Crèvecœur, 1997, 46)
Although I suspect the meaning of the term has been adjusted over the year, the basic idea that "The American" has a right to sue for their equal rights, to get what is rightly (in their eyes) theirs is profound. Whilst it has started to come over here, the fact that a main tenet of the legal system is being exported to another country through commercial interests alone tells you something about the nature of litigation.
While Europeans attempt to rally against "no win no fee" lawyers (litigation attorney's in America) and view it as something only the desperate and uncouth use, it doesn't have that tainted image to such an extent in the USA. Interestingly The Simpson's parody the typical "ambulance chasing" lawyer with Lionel Hutz, but even this says something about how prevalent, how encroached in society these law firms are.
I suspect that De Crèvecœur didn't mean the settlers go around suing each other for running a cart over his neighbours foot or firing his son for having a tattoo, you can read between the lines and trace the growth of these companies from those passages above.
DC is saying that the American is proud, fiercely protective of his own and not shy to stand up for his rights. If something is wrong, broken or unfair, he will fight to change it....not only will he fight, there are mechanisms in 18th Century America to allow him to do so. I can't imagine European workers would have had such ready access the ear of the legal system - this reflects the innovation, the new free regime and the relative prosperity of America.
Also, because everything is still new and there are riches to be had, setting out a social order is paramount. Unlike Europe where boundaries (both physical and social) have been set in stone for centuries, Americans are still find their place in the country. How this relates to the modern law firms is debatable - but certainly many of them believe they are actively helping to change society for the better.
Another point one could make about litigation and the "freshness" of America is how these firms have diversified into overseas contract law, new technology and immigration law. Possibly no other institution from the beginning of America has been on the cutting edge of societies dilemmas And unlike British or French law which is based on hundred of years of precedence and crown intervention, the American legal system is unhindered as long as it operates within the general tone of the Constitution.