First I'd like to comment a bit on that map. Maps like that pop up from time to time (I believe there are some versions of it on wikipedia as well). For anyone who is curious, it is based on the US Census Bureau's American Community Survey, which asks questions about ancestry. It should be noted that ancestry is self-reported, and respondents can pick more than one option. A copy of the most recent questionnaire is here. I should note that this is NOT the decennial census, but surveys that the Bureau does with selected respondents in off years.

As for the map (or versions of it), it's misleading in two ways. First is that it shows results by county, so a county with 50,000 people or a county with 4 million people will look the "same" as long as they're the same size. A second way these maps are misleading is that they are showing the largest self-reported ancestry out of a vast number of choices. Thus an ancestry "wins" a county just by being the biggest percentage (which out of many choices means it could be even something like 10% of total respondents!). ACS actually has a map of county results for reported German ancestry here as a pdf, which I think better contextualizes what sorts of percentages we're talking about by county. The areas where reported German ancestry actually approaches a majority on a county basis are in the upper Midwest (the Dakotas, Nebraska, Iowa, Wisconsin and Minnesota) - the rest of the "German" counties have much, much lower percentages in general. For the US as a whole we're talking about 40 million people self-reporting German ancestry, or about 13.3% of the whole. Which is big! But partially it's because other ancestries like those from the British Isles get broken down more (Irish is over 10%, English is 7%, "British" is about 1%, then there's Scottish, Scots-Irish, and 20 million people reporting "American" which often includes a lot of people with British ancestry but too distant in a way for them to really care about it). Specifically for those with ancestry from Ireland this makes it look a bit smaller, because generally those with Protestant ancestors and those with Catholic ancestors use different terms for their ancestries, while those with Protestant or Catholic ancestors from Germany will just say "German".

Another thing to notice is that German immigration to the US is a relatively old phenomenon. Much of German immigration occurred in the 1840s and 1850s, with a second "peak" in the 1870s, and then a rapid dropoff. Irish immigration did have its major peak in the 1840s, but there was a fairly steady flow of immigrants for some time afterwards (Ireland didn't actually stop being a net exporter of migrants until the 1990s), and Italian immigration was a phenomenon of the late 19th and early 20th centuries (along with other immigration patterns from South and Eastern Europe) until the doors to the US were slammed shut in the 1920s. Overall something like 5.5 million immigrants came to the US from Germany between the 1820s and 1924.

So there's some context. Now, it's worth mentioning that the German American community had a very strong identity, especially in areas where they comprised large chunks of the population, such as the Midwest (Milwaukee's and St. Louis' beer industries are a thing for this reason!). Members of these communities had their own churches, community organizations, and even German-language schools and media (something like 1,000 German language newspapers were printed in the US in the 1890s). This is why someone like Lawrence Welk (born in North Dakota in 1903) could grow up being bilingual in German and English.

So what happened? Well, a big part of what happened was the First World War. As I discuss in this answer about the darker aspects of the Wilson Administration years, there was a big upsurge in xenophobia during the World War One years, with many school districts banning German language instruction and "pro-German" textbooks. The German language press was also hit hard by the Espionage Act of 1917 (authorizing the Postmaster General to revoke mailing priveliges for "suspect" publications) and the Trading with the Enemy Act of the same year, which required all German language publications to file English translations at post offices on political news or commentary, with imprisonment and fines for noncompiance. Many German-language publications also lost advertisers in this time and folded (although I don't have an exact number of how many). There was a popular push for German Americans to assimilate from this time and onwards ("America First and America Only", and concerns about "Hyphenated Americans"). A lot of this involved switching not only to English but widespread Anglicization (one of the weirdest examples was renaming sauerkraut "Liberty Cabbage"). The German American community as an identity never really recovered from this, and assimilation continued apace from then onwards (for example, despite 40 million Americans claiming German ancestry, less than 900,000 reported actually speaking German, with another 560,000 speaking "Yiddish, Pennsylvania Dutch or other West Germanic languages").