This problem is a little tricky in a way that probably was not intended. It would make more sense if the string was replaced with a spring.
When you drop the mass, it accelerates. But the disc stays still.
When the string becomes taut, that must suddenly change. The mass and the edge of the disc suddenly start moving at the same speed.
If they were connected by a spring, you would see the spring stretch, pulling the mass and the edge of the disc towards each other. The mass would slow down and the disc would start to spin. You would understand why conservation of momentum applies.
There is another kind of interaction like this - a collision. These also involve a sudden change in speed. Typically you don't think about what happens during the collision. You just work with speeds, momentum, and energy before and after.
Treat this like a collision. When the string comes taut, there are for a moment very large forces that slow the mass and make the disc rotate. Don't worry about those forces. Instead, use the conservation laws on the before and after situations.
Treat this like an inelastic "collision". That is a collision where two things stick together and move at the same speed. Kinetic energy is not conserved. You can see the similarity here, where the mass and the edge of the disc move at the same speed afterward.
Note that if they were connected with a spring, they need not move at the same speed afterward. That would be an elastic "collision".
Another way to set this up would have been to remove the string. Move the mass above the disc and coat it with glue. Drop it. As it passes the disc, it just touches the edge and sticks. That would be the same problem, more obviously as an inelastic collision.