As you can see from this definition at Macmillan, the phrasal verb pass on has a special meaning:
to give someone something that someone else has given you
So when you pass the pepper to me, the meaning is more general. You might have received it from someone else first, or you might have been the one to the lift the pepper from the table before passing it to me.
When you pass the pepper on, you have definitely received it from someone else before sending it on to its destination. Furthermore, the ultimate recipient might be the person sitting next to the person to whom you pass the pepper, or might even be farther away.
Children in particular like to use this phrase. Imagine a bunch of them queued up for some activity. The teachers have ordered silence, but one plucky fellow whispers this to his neighbor:
Timmy loves Nancy. Pass it on!
In a few moments, everyone in the line knows the gossip.
At least that's the way it used to work. Today, I suppose, the news spreads over Twitter.
You can see why pass on is so fitting for a virus. You receive the virus from one infected person and then give it to another. Macmillan offers this example, which is spot on:
I took the day off work because I didn’t want to pass on my flu to everyone in the office.
As with many phrasal verbs, the sequence is a little flexible. I'll reorganize that example a little:
I took the day off work because I didn’t want to pass my flu on to everyone in the office.
I do not know why a comment indicates that this is wrong. Perhaps it is correct only in America.