Some subsystems have a small "head" icon above their power levels. These systems are:

• Piloting
• Shields
• Weapons
• Engines

By staffing these systems with a crew member, you increase the abilities of that system. When you send a crew member into an unstaffed room, you can see them head straight for the console and start working there.

From my observations:

• Staffing the Cockpit and the Engine Room increases your chance to dodge.
• The Engine Room also has the added benefit of reducing FTL charge time when staffed.
  • Please note that staffing the Cockpit is required for almost everything; the engines won't recharge even if someone is manning the Engine Room. Your dodge chance also goes to 0% if no one is trying to pilot the ship; it's an inert hunk of metal.
• Staffing the Shield System results in shields recharging faster.
• Staffing the Weapons System results in your weapons charging faster.
• Staffing the Door Control System results in your doors upgrading a level.
• Staffing the Sensor Control System results in your sensors also upgrading a level.

The amount of the benefit you receive is based on the crew member's system skill level, which you can see by going to Ship --> Crew, and hovering over the crew member. They will gain experience by working on the system, increasing the bonus they provide to the relevant system. You can occasionally see your crew gain experience by seeing yellow icons appear above them. These icons will match the system they are currently working on.

Note that it doesn't seem like crew gain experience when the ship is not in danger, so staying in a system doing nothing is unlikely to help your crew gain experience.
The exception to this, as @Halfwarr noted, is asteroid fields. Since your ship is continually in danger and the asteroid damage can be mitigated, this is probably an excellent way to gain skill. This is no longer valid with the full release.

Also, there is only a single console per system, no multiple crew members do not provide any bonus beyond faster repair speed in the event the system becomes damaged. This is easily testable by sending another crew member to the cockpit; the first one will provide a bonus to dodging when he reaches the console. The second will not increase your ability to dodge.

I have not unlocked all ships, however, so it may be possible that some of the later ships have multiple consoles for some or all systems, in which case, multiple crew members would provide a greater bonus.

As you noted, if it picks up Engi, fluffwise there's no reason they shouldn't pick up drones. This game doesn't try very hard to make biological sense; don't get caught up on it.

Gameplaywise, while it may feel misleading to lase an enemy to find him completely freaking unharmed, it's more misleading, and more dangerous, to teleport into a ship and be caught unaware by more bodies than were on your sensors.