There is some research about using the process (reversing new capillary growth) to fight existing tumours or cancer but even the originator of the idea dr. William Li, promoted it as a preventative, not a therapy.
The anti-angiogenic diet doesn't destroy all capillaries, it inhibits new growth and tends to restore the number of capillaries back to your bodies 'baseline'. So it differentiates between new growth and existing capillaries.
Since fat is fed by capillaries, the removal of these capillaries will starve the fat cells, and you will lose fat - that is the general idea.
Here is Dr. William Li's Ted Talk:
https://www.ted.com/talks/william_li_ca ... rve_cancer
Here is his list of foods that he says inhibit or reverse new capillary growth:
