That has mainly been the debate in the United States due to the political tradition of people such as Thomas Jefferson, the rest of the word had a very different historical tradition of governance.

To get into that, we first have to get into what the Left and the Right is and then how the left developed.

The Left and the Right come from the French Revolution, where the people who sat to the left side of the Estates General were more radical while those who sat to the right were more moderate or reactionary. This term has been adopted globally and has been used to divide politics ever since.

What we had at the time of the French Revolution developed into illiberal Leftism, as people on both sides purged their political enemies and for a while a fair number of people on the Left tried to get rid of Christianity in France and worship Liberty itself as some sort of god (I did find them to have pretty sound reasoning for it, if you'd like me to expand upon it).

With this in mind, as the Industrial Revolution began and then went into full swing, workers began advocating for government more representative of their wants and needs, and even governance by then, which is how communist and socialist ideologies developed. Funny enough, these ideas became associated with the Left because of the Second French Revolution, where left wing Republicans went up against the right wing French Monarchy. This was also the time where the Paris Commune came to exist, for a short period of history, which strongly influenced much communist and anarchist thought.

Throughout the rest of the 19th century, Left Wing politics were divided between socialists and liberals, neither of which generally worked with the other, but both wanting some of the same things (Liberals at the time were very much in favor of Free Trade, which Socialists and Communists generally opposed).

Environmentalism largely developed as an ideology in the United States during the 50s and 60s, seemingly joining the Left because that's where the rest of their political beliefs aligned. Around the same time in Europe, social democracy developed as an idea, which combined socialist and liberal positions, partially as a counter to the USSR's authoritarian communism.

Edit: /u/Cosmic_Charlie has a much better summary on Environmentalism in response to this comment.

Anarchism, while existing and being a pretty fascinating ideology, never really got off the ground outside of Spain, due to those who would support anarchism already being socialist or communist (at least according to Bookchin, he claims that an Italian anarchist being one of the first major thinkers to come to Spain to speak to workers was the reason they developed an anarchist ideology as opposed to a communist or socialist one).

If you would like me to expand on anything, let me know.

Sources:

Mike Duncan's Revolutions Podcast, Part 3 on the French Revolution

Murray Bookchin's To Remember Spain: The Anarchist and Syndicalist Revolution of 1936 and Post-Scarcity Anarchism

Edit: Spelling, sorry I was on mobile before.

As BSebor has talked about France, I'll talk a bit about Britain.

Britain developed two main political groups, the Tories (pro-Monarchy) and Whigs (anti-Monarchy). Over time, these developed into:

A)The Conservative party: a party of traditional ideas like Anglicanism, the aristocracy, Empire, and protectionism. The Tories (note we still often refer to them by their old name) were more popular with the land-owning gentry and rural farmers.

B)The Liberal party (originally an alliance between Whigs, pro-free trade ex-Tories, and radical non-conformist, meaning non-Anglican Christians): more popular with the new industrialist middle classes, and supported low tax, free trade, and religious tolerance (well, primarily denominational tolerance, but they were in favour of tolerance towards Jews as well).

Now, for most of the 19th century, the poor couldn't vote, but that didn't mean the working class just ignored politics. In the early part of the century, the Chartist movement lobbied for rights and worker protections, and although it faded away, the latter half of the century saw the formation of Trade Unions, which fought for the rights of the Working Class. While many rural farmers voted for the traditionalist Tories, once they got the vote near the end of the decade, the upper working class (such as tradesman) in the cities normally voted for the Liberals (though not always). Another point that should be mentioned were the evangelicals, who became very prominent in the latter half of the 19th century. They wielded signifcant influence in parliament. While they weren't primarily aiming at improving worker's rights and pay, many of the moral issues they pushed did improve life for the poor. For example, their concern about young girls and boys working together in mines with little clothing (it might lead to naughtiness!) led them to push for child labour laws.

However, there was no party directly representing the rights of the working class. So, a number of socialists and union leaders created the Labour party to improve the pay and conditions of the poor. Throughout the early half of the 20th century, it gained more and more seats in parliament, replacing the Liberal party as one of the two main parties in Britain.

Now, as Britain had absorbed the terminology used in France, the Conservatives were referred to as right-wing, while Labour was referred to as left-wing. Thus, you can see that it's a bit odd to say "when and how did concepts such as... taking care of poor people come to be associated with left-wing politics" because for British leftists, taking care of poor people was the fundamental main point of left-wing politics. If a party didn't advocate taking care of the poor, then it was hardly left-wing.

Now, as to racial equality, there are three main reasons that became associated with the left. Firstly, because a number of Conservative politicians were against it (on a rhetorical level, and in terms of Imperialism: it's important to note that, unlike America, Britain never had segregationalist laws). Secondly, because of Hitler. For left-wing activists, Fascism was the ultimate enemy (and the favour was returned), and the Nazis were defined by their hatred of other races, thus left-wing politicians would be proud of their racial tolerance by contrast.* However, the main reason is one that goes back to the founding roots of both liberalism and socialism, the two main ideologies in British leftism. Liberalism's focus on defending the rights of the religiously oppressed, and Socialism's focus on defending the rights of the economically oppressed, led to many left-wing academics criticising Britain's empire, and thus the racial ideology that painted "ignorant" blacks or Asians as "savages", that underpinned it.

*It's important to say that Labour politicians in the 50s and 60s were not as anti-Empire or pro-racial equality as their successors in the 80s and 90s would have liked to claim they had been. They were, after all, products of their time. But they weren't as overtly racist as Conservative politicians like, say, Enoch Powell, and never used fears of immigrants as a way of getting votes.

Consider the 1964 election for Smethwick. The Labour MP had voted against the Commonwealth Immigration Act that sought to reduce immigration to Britain (primarily by Afro-Caribbeans). The Tory candidate, Peter Griffiths, ran and won on the slogan, "If you want a n****r for a neighbour, vote Labour." But without the asterisks, of course.

As to environmentalism... I'm going to be honest, I don't know how that became associated with the Left in Britain.

I will make one final note, in regards to your sub-title: "I always thought the issue was more about whether or not the government played a role, not whether these are actually good concepts."

It's important to remember that, in terms of Britain at least (though I'd say it applies to many other countries too), the formation of "Right-Wing" and "Left-Wing" had very little to do with how significant the role of government was. In the 19th century, the Right were aristocrats, the high clergy, the landowners, the army, and the pro-Empire establishment. The Left was liberals (who of course, were primarily interested in Liberty, and thus small government), and some anarchists and socialists (and remember, this is before the USSR, so socialism wasn't really associated with dictatorship or totalitarian governments).

Small government as left-wing and big government as right-wing was something that developed with a)the rise to prominence in the Labour party of those who advocated centralised, state-run industries as the best way to ensure the economy served the people, during WW2 and afterwards (though there were people advocating other ideas within the party), and b)Thatcher's revolution within the Conservative party, turning them from the party of tradition to the party of the radical free market, and thus privatisation, adopting ideas traditionally associated with the now-tiny Liberal party.

The idea that Left means big government and Right means small government is one that British people have now largely absorbed, but it's not universal. There's plenty of people on the Left who would disagree with the idea.

Although I'm not sure when racial equality first became associated with the "left," the the association is a long one. Even prior to Marx, socialism was seen to favor racial equality (and certainly abolitionism at a minimum). For example, George Fitzhugh defended slavery from what he certainly viewed as left-wing attacks in 1854 in the book Sociology for the South. I know this is a long quote, but it does serve to give the flavor of the overall argument, and it actually sounds rather modern in its slippery-slope condemnation of equality as a value:

“The Abolitionists themselves furnish the most conclusive evidence that slavery must exist in every society until human nature itself is changed. Nay, they propose to change all man's nature, in order to fit him for that social equality, that community of property, and of other things more sacred than property, which they would erect on the ruins of our present system of society. The [abolitionists] hate slavery, and seeing that marriage brings about one of the forms of slavery, to be consistent, they will have no more marriages after the old fashion. Separate property, too, gives power to those who hold property to command the labor of those who hold none. ‘Property,’ say they, ‘is a thief!’ and must be abolished. The Bible commands wives to obey their husbands, and slaves their masters; the Bible must be cast into the flames! Christianity and Socialism are deadly enemies. But after all the institutions of society are destroyed, families abolished, churches demolished, the Bible burnt, and property held in common, still they have the candor to admit that the selfishness of human nature would for a time disturb the harmonious working of their system. They promise us, however, that a few generations would change and perfect man's nature, and then Socialism would work admirably. At the end of the time we suspect they would become converts to the sage reflection of Christopher North: ‘There is a great deal of human nature in man!’ We treat the Abolitionists and Socialists as identical, because they are notoriously the same people, employing the same arguments and bent on the same schemes. Abolition is the first step in Socialism: the former proposes to abolish Negro slavery, the latter all kinds of slavery, religion, government, marriage, families, property, nay, human nature itself. Yet the former contains the germ of the latter, and very soon ripens into it; Abolition is Socialism in its infancy.”