Technically speaking, “Anglicisation” refers to the process within English when we borrow words from other languages and, well, assimilate them by making them sound a bit more English. This also is used to refer to the reverse process by which other languages borrow English words, the individual words being more precisely referred to as “Anglicism.” While this back-and-forth has been a part of the development of language since we first started communicating with each other, in the last century it has exploded with the global popularity of American and British media and culture.
This has been especially conspicuous in German, a language closely linked to English both linguistically and culturally. They are both derived from the Germanic language, and have evolved largely independently since then. Thus, It should be noted that not all similarities between English and German are due to an Anglicizing influence on their culture, as many existed well before then.
English has experienced its own influxes of other languages, most notably French, with a smattering of other tongues, just as German is experiencing currently.
Of course, English isn't the only culprit here. Speckles of French, Russian, and Italian may also be seen, especially at a regional level where there was once some invasion or occupation or another, like in the German dialects of Alsace-Lorraine or Berlin (Berlinerisch.)
It should also be noted that while most German speakers have at least an intermediate understanding of English, required to a fairly advanced levels within the education system, proportionally few English speakers know anything of German. Other than the odd song and (translated) novel, German culture has had remarkably little influence on American at large, whereas American culture is all the rage in the malls and movie theaters of Germany. Also, many international businesses within Germany simply conduct all of their affairs, internal or external, in English. Similarly, international companies with a presence in Germany often do not even bother to translate their products in German, leaving them in English. A certain amount of mixing of the languages under these circumstances is hardly surprising.
So, where can we find specific examples of this effect reflected within German itself? Here's a few: