Why does Liberalism have a different meaning in the United States than in Europe? When did 'liberal' become synonymous with center-left politics?

So, this is a very complex question, to the point where entire books could be, and likely have been, written on the question of the development of Liberalism in the 19th and 20th centuries. You are also asking two different questions here, why is there a difference between continental usages of the term, and why Liberalism became synonymous with the center-left.

I will tackle the latter, rather than the former, as the specific reasons for etymological terms being utilized differently in different areas isn't my area of expertise. Also, for anyone looking to put all this in its proper context, and he does a masterful job of this, I'd suggest Eric Hobsbawm's Age of series, where he charts historical developments, including the development of the ideology of Liberalism, throughout the so-called "Long Nineteenth Century" between 1789 and 1914.

I will be simplifying here greatly for the sake of brevity and specificity. Liberalism is a political ideology that emphasizes the liberty of the individual, although what that means in practice is constantly changing and evolving.

A prime figure in Liberalism is John Locke, who wrote that individuals had natural rights to life, liberty, and property, and that it was unjust of governments to arbitrarily seize or restrict these natural rights. In modern times these are often called Classical Liberals.

However, there is also another political philosophy growing out of Liberalism. This philosophy is often called Social Liberalism. Many Liberals began believing that the promises of liberty and equality of Liberalism couldn't be achieved without the amelioration of material conditions and market intervention, something that Classical Liberals abhor. Classical Liberals believe that all people working in concert towards their own happiness, specifically through a free market. There are many shades of grey between these two poles, but that is the basic difference between the two different camps.

Classical Liberals believe a free and unregulated market, alongside the rule of law and democracy, are the best tools for maximizing human happiness. Social Liberals, on the other hand, believe that social conditions need to be ameliorated through government action and that a totally unregulated market can be harmful to the human condition. Many Classical Liberals made alliances of convenience or ideology with Conservatives as time went on, while Social Liberals carved out a middle position between the Conservatives and the rising Socialists, and in the modern age it can be hard to distinguish the two.

A good figure to illustrate this trend is John Stuart Mill. Another influential Liberal thinker, Mill already shows the beginnings of the move towards a more Social form of Liberalism in his earlier years.

He passionately defends the rights of the individual to be free from the tyranny of the state and that intervention by the government should only take place when an individual is harming others. He considers progressive taxes to be a form of mild robbery upon those who work harder. However, Mill believed that in theory, if it could be justified on utilitarian grounds, that is increasing the overall amount of happiness the most, government intervention could be justified.

However, later on, Mill trends more and more towards intervention in the economy and economic justice. He championed worker's co-operatives and believed that the capitalist system hindered the continued development and happiness of humanity. The object here has not changed. He championed the freedom and equality of humanity in both cases. To simplify his work, and Liberalism's changes in general greatly, the ends did not really change, only the means and the scope.

If anyone has any questions or requests for clarification feel free to ask, as this is a very brief overview of a very complex topic.