Why is there a mass limit on biological powered flight?
One of the physical limits to biological flight is muscle physiology.

Muscle force output is proportional to muscle physiological cross sectional area (PCSA) multiplied by its specific tension (Gans, 1982):

$$F = \text{PCSA} \times \text{Specific Tension}$$

PCSA is basically just the cross-sectional area of a muscle adjusted for its architecture. Pennate muscles have higher PCSA than parallel fibered muscles, because more fibers are packed into the area (at the cost of shorter functional distances). Specific tension is the force output per unit area. For a single species, specific tension is essentially constant.

So the way to make more force is to make more PCSA. But muscles don't scale only in area, they scale in volume as well. Increased volume increases mass as a cubic power, but muscle area only scales as a square power. So any increase in area that requires a volume increase (which is inevitable), will quickly outstrip any gains in force output.

Note that this doesn't mean that very large flying vertebrates were not possible: the largest pterosaurs (e.g., Quetzalcoatlus) had ~15 m wingspan and perhaps weighed 200 kg and the largest birds (e.g., Argentavis) had ~7 m wingspan with a mass over 50 kg. Both, however, were likely gliders.

Gans C. 1982. Fiber architecture and muscle function. Exerc Sport Sci Rev 10:160–207.