Despite its name, a Low German speaker shares more foundational sound patterns with an English or Dutch speaker than with someone speaking Standard High German.
To understand the gap between the two, one must look at the High German Consonant Shift. Between the 3rd and 8th centuries, a massive phonetic change swept through the southern and central parts of the Germanic-speaking world. In these mountainous "high" regions, hard consonants softened or shifted. A "t" sound shifted to an "s" or "z" sound, turning the word water into Wasser. A "p" became a "pf" or "f", turning Appel into Apfel, and a "k" shifted into a "ch", changing maken (to make) into machen.
The northern, low-lying coastal areas never adopted this shift. Because the people there kept the older consonants, their speech remained closely aligned with the Anglo-Frisian and Franconian languages that eventually became English and Dutch. Strictly speaking from a linguistic standpoint, Low German (Plattdeutsch) and High German are distinct branches of the West Germanic language tree.
However, before the era of modern borders and standardized education, Western Europe existed as a vast dialect continuum. A traveler walking from the Alps to the North Sea would notice local speech changing slightly from village to village. Neighbors could always understand each other, even if people at opposite ends of the trade routes could not communicate at all.
The reason Low German is categorized under the "German" umbrella today comes down to a cultural and political shift. During the Middle Ages, Low German was the powerful, international prestige language of the Hanseatic League, dominating trade across the Baltic. But in the 16th century, Martin Luther translated the Bible using Central and High German dialects. As the printing press distributed this text far and wide, Standard High German emerged as the dominant written language across the fractured states of the Holy Roman Empire.
When Germany eventually unified into a single nation-state in the 19th century, Standard High German was entrenched as the language of government, education, and media. Over time, Low German was administratively and socially demoted from a distinct regional language into a mere "dialect" of the German nation, despite its unique linguistic origins.