Why are my crew starstruck?

It means the crew has been stunned. The effects are only temporary though.

There are a few sources of stun but all of them will only be found in the Advanced Edition (AE).

• Stun bombs, bomb-weapons that deal 1 system damage, in addition to stunning crew in a room for 15 seconds with a 100% chance.
• Ion stunners, ion-weapons that fire a yellowish particle that, if it hits, deals 1 ion damage and stuns crew in the targeted room (or shield room if shields were hit) for 5 seconds.
• Hacking drones that have been activated coming from a ship with the "Hacking Stun" augment will cause stun to all foes in the room or that enter the room until the hacking is on cooldown. (This augment costs 60 scrap and is not found on hostile ships.)
• Ion Intruder, a boarding drone that deals ion to systems will stun crew for 5 seconds if they are in the room being ionized.
• Heavy Missile weaponry/Heavy lasers now have a chance to cause stun for a few (5?) seconds. This is a low probability side effect added to existing (non-AE) missile/laser weapons. The probability is 20% for such lasers, unknown for missiles.

All of these weapons will cause the yellow stars and an inability to work or move. The weapon-based stuns will stun friendlies as well as hostiles that are in the blast radius.