How do we actually know that no other people had been living on American soil other than the well known Native American Indians

Every ancient human remain found thus far in the Americas, including the very earliest like Anzick-1 and Naia (both around 12,000 years old) has been shown to have genetic affinities with modern Native American populations. There is no reliable evidence that indicates that an unrelated population ever settled the Americas south of the Arctic (modern Inuit populations weren't part of the initial wave of human settlement in the Americas, arriving as part of a more recent expansion of Siberian peoples around 4500 years ago).

where did those Native American Indians that are well known come from?

Except for the aforementioned Inuit and related cultures (Yupik, Aleut, & Inupiat), other indigneous peoples have had ancestors in the Americas for at least 16,000 years and likely closer to 20,000 years. By 23,000 years ago, the ancestors of the First Americans were already a genetically isolated population, but we don't know if they were already in the Americas at that time or if they were in some remote part of Eurasia. Regardless, not long after, they crossed into the Americas via northeast Eurasia. While an overland route had been favored by older generations of anthropologists and archaeologists, currently a coastal route appears more likely for the initial wave, perhaps with an overland expansion following once the Ice Age started to thaw. Over those 16-20,000 years, Native cultures diversified and developed into the ones present at the time of European contact and beyond.