Usually I pick a few of the nicest, longest, seed pods once they start to shed their outer green fleshy coat but before the hard woody pod starts to split so I make sure I do not lose those seeds if they split open early and spill all over the ground. I keep the pods in a concrete building of ours outdoors and since they usually start to reach this stage in fall the building is rather cool as they finish drying and finally split open on their own then I can pick all the seeds out.
If I have a lot of smaller pods that are not as impressive to collect for next year or some late season pods I sometimes even leave them on the plant until really late fall after frost kills the plant and then throw the pods in a bucket in the building to pick seeds out whenever I have the time once they have finished curing.
Either way, the seeds don't seem to matter which way I harvest them as long as they are mature and no longer increasing in size. I just like to wait until they shed the green coat so I am sure they are completely mature, especially if the pod is a really evil looking large one so I am sure to save viable seeds from that wicked pod for next year.
Once I have separated the seeds I usually put them in a zip-lock baggie and store them in the cool basement with our other garden seeds like pumpkins, watermelons, etc. I have never tried storing them in the fridge, I think my family would have been less than pleased with that. I don't know if storing them in the fridge would increase their germination, although I have never had much of a problem with germination and the storage method I have used so far since I keep them cool and dry in the basement.
I don't know how much digging out of seeds you have done yet already but make sure you dig all of them out. There are some in the hollow cavity between the 4 prongs that can be easily shaken out or easily poked free but then there are alot more hidden behind tissue layers along the outer walls of the pod. I like to use a pokect knife to dig those out ( but then again that just may be because I like to play with sharp pointy things... ) Not like you NEED to get each and every seed because you won't have many based on the amount and size of your plants, but otherwise if you go to compost the seed pods or whatever they will reseed ALL OVER and your entire gardens will become infested with them.
Another thing to note is that I often separate seeds into "perfect, normal creamy to brown" and "kinda off-color gray" sets and store them separately in case the grey "maybe moldy ones" actually did have mold issues and would spread and cause problems for all the seeds. Apparently Crissytal, who also got some seeds, has a problem with some black and blue speckles which may be mold on some of her seeds when she has been harvesting some of hers. So that is just something else to possibly keep in mind when harvesting seeds.