Does cooking destroy the nutrients in food?
Some, sometimes — and it increases the availability of others, which is the part usually left out. Whether cooking is a net loss or gain depends entirely on the nutrient and the method.
What cooking genuinely reduces:
Vitamin C, which is both water-soluble and heat-sensitive, so it is the most vulnerable nutrient by a distance. Boiling loses the most, because it is lost to the water as well as degraded.
B vitamins, particularly folate and thiamine, water-soluble and moderately heat-sensitive.
Some polyphenols, though results vary.
What cooking increases:
Lycopene in tomatoes becomes substantially more available with heat, because cell walls break down. Cooked tomatoes deliver more than raw ones.
Beta-carotene in carrots and similar vegetables, for the same reason.
Protein digestibility improves with cooking, as does starch digestibility.
Iron and zinc availability improves where cooking, soaking or fermenting reduces phytates.
Toxins and anti-nutrients are destroyed. This is not a minor footnote: kidney beans are genuinely dangerous undercooked, and several staples require cooking to be safe or usable at all.
What the method changes more than the fact of cooking:
Boiling loses the most water-soluble vitamins — much of it into the water, which is why using the cooking liquid recovers them.
Steaming retains considerably more.
Microwaving performs well, contrary to its reputation, because cooking times are short and little or no water is used.
Roasting and grilling retain water-soluble vitamins but involve longer exposure and higher temperatures.
Cut size and surface area matter, so cooking vegetables whole retains more.
Time matters more than temperature for most losses.
The perspective that matters most: losses are typically in the range of tens of percent for the most vulnerable nutrients, and eating more vegetables in a form you enjoy comfortably outweighs the difference. Frozen vegetables, blanched and frozen soon after harvest, frequently contain more vitamin C than "fresh" produce stored for a week.
General information, not dietary advice.