Why are restriction sites palindromic in nature?
Not all restriction sites are palindromes : https://en.wikipedia.org/wiki/List_of_restriction_enzyme_cutting_sites:_E%E2%80%93F#Whole_list_navigation

EcoRI: GAATTC (note that this is not 'palindromic' in the usual sense (e.g.'anna', GAATTC read backwards actually gives CTTAAG), but only if given the G-C A-T binding)

The double strand of DNA is composed of one strand in direction 5'-3' and another into the other direction (3'-5'). Restriction enzymes tend to be homodimers, thus would bind to both sides of the double strand, expecting the same recognition sequence. But as one strand runs into the other direction, the strands need to be palindromic to form a double strand.