Is what my mom said on cats correct?

She's partially correct. It's also the fact that in the wild cats are both prey and predator. A sickly one will attract predators to the colony, and put them all in danger. They drive off the weak and sick to keep the rest of the colony safe. The "smell" that the ill cat puts off is very upsetting to your other one. It may be best to keep them separate until the other one is better, or the aggression that is being shown may continue after the first one is better.