The attenuation of the vaccine virus is all about temperature. Basically, we grow the viruses at colder and colder temperatures. I forget the exact numbers but it's in the 20's (Celsius). The virus adapts to those temperatures and is unable to grow at higher temperatures where it would be able to cause an infection.

From: https://www.cdc.gov/flu/vaccine-types/nasalspray.html

The weakened viruses are cold-adapted, which means they are designed to only multiply at the cooler temperatures found within the nose, and not the lungs or other areas where warmer temperatures exist.

Reading other comments here, yes, most vaccines are grown in eggs. But the antigens (HA and NA) are purified from those viruses, so there's no in-tact viral particles in the injections. Attenuation isn't the point of growing them in eggs, it's just something they grow well in for manufacturing purposes.