What do the reputation percentage values in the System Panel mean

As cmdr Freiheit posited in comments, the integrity rating is an indication of how much damage a component can take before failing.

This was stated by Frontier Developments staff on the game's forum here: https://forums.frontier.co.uk/showthread.php?t=102717&p=1591966&viewfull=1#post1591966

"Also the [specific variant of module] has a higher integrity rating so that it can take more punishment before failing."

Basically, they're asking you to be a smuggler.

Ultimately it will mean two things while you're carrying that cargo:

1. You will need to do a sneaky station entry, unless it's an outpost.
2. You will need to avoid being scanned by any system authority ships, such as if one interdicts you.

If a system authority ship (including those orbiting a station entrance) does the kind of scan that shows up as a "scan alert" while you're carrying the illegal cargo, you'll get a fine. If you don't pay the fine within 24 hours it will turn into a bounty (and you'll be wanted in that system). Being caught with illegal cargo will give you a bad reputation with the local system authority faction.

If you're unlucky enough to get caught with the illegal cargo in multiple systems, you could end up with multiple fines. I believe if you're caught again by the same authority (interdicted and scanned then enter a station they control), you won't actually get fined a second time for the same illegal cargo containers.

Note that some missions that don't have that warning, are actually for commodities that might be illegal in other systems.