What does it mean to “re-tank” toilet

I believe that you are correct and that the container was added to reduce the amount of water used per flush. As for whether it will affect your fix - it might. If the tank is leaking then unless it is cracked then there are only two places that it could be leaking from - where the inlet connects to the tank or where the tank connects to the bowl. If it's leaking from the inlet valve then the container would have no effect other than giving you less space to work. If it's leaking form the spud washer (the rubber gasket between the tank and bowl) then you may need to remove it to get to the bolts that hold the tank to the bowl (though it appears that the container has indentations that go around the bolt heads so again it may just come down to the only effect being a more cramped space to work).

That's a fairly popular fill valve. To open it:

1. Shutoff the water, you don't want a geyser.
2. Lift the float to the top position.
3. While holding up the float, push down on the top cap and twist the cap a quarter turn (I believe counter clockwise).

The cap will then lift off and you can inspect and clean any parts. Before reinstalling the cap, move any small parts to a safe location, get a small cup to place over the top of the valve, and turn on the water supply. This will jet water straight up from the top of the valve and clear out any debris.

Reinstalling the cap is the reverse of the above procedure. Put the parts back, position the cap back on top, lift the float, press down on the cap, and twist.

Note that if cleaning it doesn't work, it's fairly simple to replace this fill valve.