The simplest explanation is just to wave your hand and say, "Space magic!"
A more complicated explanation is that the asari reproductive process enables their bodies to manually randomise the DNA they pass on to their offspring, similar to the way some species on Earth such as jellyfish and flatworms can adjust the composition of their own cells. I actually suspect that their reproductive process has nothing at all to do with asari biotic powers; it's a purely biological process.
In fact, I'm constantly puzzled over HOW exactly all asari are natural biotics. Normally, biotics are created when a fetus is exposed to dust-form element zero, which suggests that Thessia may have been a planet naturally rich in element zero, but that doesn't explain how asari born on planets other than Thessia (and presumably without exposure to eezo accidents) still end up as biotics anyway.
That suggests to me that either:
a) All asari possess a natural reserve of dust-form eezo in their bodies, and a portion of this is passed on to their offspring. However, this would also mean that each new generation of asari born outside Thessia would get progressively less and less eezo, which means gradually decreasing biotic prowess.

Asari are the only race capable of producing either eezo, or some biological substitute, within their own bodies. In which case, I'd sure as hell have my scientists trying to figure out what exactly this substance was, and whether it can be manufactured in a lab!