I have Shimano PD-T780 pedals. I bought them almost new only a year ago and they aren't cheap pedals.
The axle thing has a hex indentation on the side not visible in the picture which is used to screw it into the crank. The silver wrench ring (like a nut but no screw threads visible) attached to the platform is labelled "Tighten 10Nm" and I haven't ever turned it. I don't know what it's for. Looking into the platform, I think bearing balls were visible. Maybe it's designed to come apart like this for servicing.
Here is an exploded view: I can't seem to upload the picture I extracted from this pdf. The two leftmost pieces of assembly 1 (the ring is rubber, a seal I guess) are on the crank, the other pieces are in the platform.
I was in something of a hurry, so I just stuck it back on and used another bicycle.
How do I properly reassemble this? And while I'm at it, should I add some grease?