Consider the formation of the gametes (the sperm and the egg) in each of your parents. Each parent obviously has one set of chromsomes from the mother and another from their father. For each parent, only half of their overall genetic material is carried through to a particular gamete, and this could be a random combination of any of their parent's (your grandparent's) chromosomes (for example, chromosome 1 from the parent's mother, chromosome 2 from the father, etc). This occurs because the chromome pairs first replicate (doubling overall genetic material), line up randomly in their pairs, and are then split up into 2 daughter cells. The chromsomes in those cells can line up again and the sister chromatids of each chromsome can be split up into 4 gametes (each with half the genetic material of the original parent cell). This is an oversimplification of the process of meiosis So different chromsomes are inherited from each gamete, and different gametes combining in fertilisation leads to diversity. Also, fertilisation restores chromsome numbers (2 halves make a whole)

Also, when the homologous chromosomes (the chromosome pairs) line up in metaphase during gamete formation, they can undergo a process known as homologous recombination. In this entirely random process, sections of chromosomes are swapped between the homologous chromosomes. This means some of the paternal dna could be found on a maternal chromosome, and vice versa. Thus, different DNA itself is found on the same chromosome, which adds to diversity.

The randomisation of these events is further amplified when you consider that its happening in both parents, so when sperm and egg combine, there are many different genetic combinations possible from just two parents.

Thus, siblings look so different because they may inherit different chromosomes from each parent, with different combinations of DNA thanks to recombination. It's super super unlikely two siblings will inherit the exact same combination of DNA on their chromosomes and so genetic variety is allowed. This is also really important for species to survive in the long term as it gives us good diversity to handle different threats.