Basically, the pod was a small doohickey with a guy inside that's deployed to take readings of any conveniently passing ion storms. This is dangerous to the ship for two reasons: when it's deployed, the shields can't be raised, putting the ship in greater danger than it would be normally, and because the pod is physically attached to the ship, there is added stress at the point of connection. If that connection breaks, we could be talking severe damage to the ship, including having the pod slammed into the hull, hence why it has to be jettisoned when things get too hairy.
I suppose the 1960's equivalent would be a B-52 going up into a hurricane with some sort of sensor drone with a guy inside, hanging out of the bomb bay or attached to the wing, and the pilot having to dump it when the turbulence gets too heavy; the inherent drama being whether or not the guy in the drone can get back inside the plane before the drone gets cut loose.
As for why it has to be manned, as mentioned above, the nature of ion storms makes remote operation and monitoring of the pod problematic at best.
And why Finney and not Ensign Ricky? Nobody in the military has just one job. You've got your primary occupation (in my case while in the Air Force, Administration Specialist, in Finney's case, records officer), your "war skill" (what you do when the shooting starts; I worked in Rapid Runway Repair, and learned all kinds of nifty ways to patch up a torn up runway and about the various types of bombs that are used to tear up the aforementioned runway...who knows what Finney did when the order came to assume battle stations), and apparently in Starfleet, when you're on a ship on a scientific mission, you occasionally have scientific duties to perform, like manning the ion pod when it's your turn.
And don't worry, Ensign Ricky's name is somewhere on that duty roster, too. It was just Finney's turn.