Part 1/2: As best I can tell, usage of neoliberalism as a political term comes out of the post-Marxist tradition. To them:

†: neoliberalism is a coordinated but decentralized effort to privatize and deregulate as a way to subvert or replace democracy and preserve the power of elites

This is my take on their take, so I apologize if I've made any mistake. I'll come back to this, but there's a bunch of background information that needs to be in place to really dig into that claim.

They generally trace its roots to the Colloque Walter Lippman, a portion of its guests went on to found the so-called Chicago School, the Austrian School, and the German School which is also called ordoliberalism. The next organizational touch-stone is The Mont-Pelerin Society which still exists.

The usual book people read is David Harvey's A Brief History Of Neoliberalism, but based on what I've read of it I wouldn't recommend it. It honestly seems to be a polemical piece rather than a serious academic work. Philip Mirowski, an economic historian who essentially specializes in studying this topic, points out two easily avoidable errors in this article Neoliberalism: The Movement That Dare Not Speak Its Name. There are other errors, and from what I've seen there are also numerous misleading over-simplifications and hypotheses presented as facts. If you want a Left-wing perspective on this, I would strongly recommend you read Mirowski's work, not Harvey's. If you're going to read Harvey's I recommend reading it critically.

So now let's kind of go through the major points of the rationale. This is actually itself an over-simplification, but you're really going to have to do some reading to get a better idea. Now put on your Marxian hat and let's jump in:

1. Keynesianism and the post-war welfare capitalism in the US and Europe were an attempt to compromise to shore up Western nations against the internal contradictions of capitalism,

2. The turn away from Keynesianism in the late seventies was directed by the elites so they could keep their wealth,

3. The economic reforms in the late 70s/early 80s in the US and UK involved a lot of privatization and deregulation,

4. The Chicago School and Austrian School are staunchly in favor of privatization and deregulation,

4.5) Moreover the Chicago School is really just a friendlier faced version of the Austrian School,

5. The turn away from Keynesianism was heavily and personally influenced by people in the Chicago School such as Milton Friedman,

6. Milton Friedman and Friedrich Hayek (of the Austrian School) made negative comments about democracy, and privatization is inherently undemocratic,

7. Free trade is a way to subvert restrictions on "international capital" which effectively holds developing nations hostage, thus subverting democracy,

thus we conclude †. The issue is that these points are ultimately narratives about history, and while they all have facts in their favor they all also have facts that work against them. There's really a lot to be said here, but I'll just restrict myself to this: (1) and (2) are really debatable. Here's a nice post in /r/askeconomics on the stagflation of the 1970s. To summarize: Keynesianism is not the same as welfare programs and spending, and the turn away from Keynesianism is both a real response to actual economic problems the various countries were facing at the time, and overstated because the working parts of Keynesianism still live in so-called Orthodox Economics.

So let's go to your other questions:

How has neoliberalism influenced American (or European) politics, both at home and abroad?

Well, it's tough to answer. If neoliberalism just means "privatization, deregulation, and free trade" then it's influenced American politics a lot. If it means the Chicago School and Austrian School then I think its influence is overstated. Usually Reagan and Thatcher are held up as the conservative faces of neoliberal politics (at least in the West), and Bill Clinton and Tony Blair as the liberal faces (again in the West). I think the further you dig into this the less satisfying it is though. Reaganomics largely did not follow Milton Friedman's ideas, despite Friedman being an unofficial economic advisor. Friedman's support was primarily due to lowering taxes, getting inflation under control(I don't like Reagan but I will say that it is extremely difficult to politically sell a fix that is very painful in the short term, and that is a genuine accomplishment of his), and cutting regulations which Friedman was generally in favor of. Thatcher is often connected to Friedman, but this is dubious and there is a paper cheekily titled "Friedman’s Lack of influence on British Economic policy". I don't think there is a good case that Bill Clinton or Blair were particularly influenced by the Chicago School much less the Austrian School. Instead, they are called neoliberal because they are generally pro-free trade and not strongly against privatization. This barely sneaks by the 20 year rule so I'll leave it at that.

How has it affected American history? What are some of the tangible economic outcomes of its embrace?

I'd say that the two major effects have been: steadily increasing individual incomes for almost everyone but also increasing inequality. For the US, here's real median personal income, which is the amount of money that 50% of individuals earn more than and 50% earn less than adjusted for inflation. Here's one measure of income inequality for the big and rich anglophone nations. For Britain there's a lovely graph of after tax income by percentile for the last 20 years (sorry mods) which shows that pretty much every individual has been gaining although the wealthier are proportionally gaining more.

Finally, what I haven't really touched on is "neoliberalism" in the Global South. That really gets into the weeds with criticisms of the IMF, economic opinions regarding trade and capital controls, US foreign policy, the "entrepreneurial self", and numerous other complex topics from a variety of specialized academic traditions. Going by Google Scholar citation counts, "Neoliberalism: A Critical Reader" by Saad-Filho and Johnson seems to be the most cited academic survey, however looking at its Table of Contents it seems to solely look at the economics. If you want to dig into the sociological perspective, "The Handbook of Neoliberalism" edited by Springer, Birch, and MacLeavy seems quite relevant.