It would help if you could describe the insects better, and whether you see them just in the soil or on the plant, or both.
-Don

If they are in the soil it's possibly soil mites or springtails.

Maybe aphids if on the leaves.

The plant may be a goner, with all the yellow & black, but what I would do is take the plant out of the soil and rinse it thoroughly. Then I would use another pot, with fresh soil media, and repot. Get rid of the old setup entirely. Then wait it out to see if it had enough life in it to recover. I would also strongly consider buy a new plant and start afresh.

The unfortunate thing about a VFT is that while most carnivorous plants will do well enough inside, at a window sill or under artifial lighting, this does not make a good indoor plant. Neither do American pitcher plants (Sarracenia). They really need to be outside, like a Sheryl Crow song, soaking up the sun. Otherwise they grow weakly as time goes on. They may be one of the more interesting of CP's, due to its trapping mechanism and widely tissue cultured, but they are far from being a good beginner plant. It depends what you are into, but there are many sundews, bladderworts, and butterworts that do well at window sills, with minimal care.

I agree with Petiolaris on repotting. When you repot, check the roots for damage and rotting. If the rhizome, the bulb like part that the leaves grow from, is black or brown and mushy, the plant is experiencing root infection and rotting. Use Consan or Physan antifungal products to kill the fungus and bacteria, cut off the rotten sections, and repot the plant in new peat moss and silica sand or perlite in a 50/50 mix. Make sure none of the mix has fertilizer, like the small bags of moss usually do. Use only dry bales of moss sold in 2 foot plastic wrapped cubes. If the plant has any life left it might recover.

If the rhizome is white and/or red colored and firm, it is at least not rotting. Use a water based insecticide like pyrethrines, malathion, ortho products and such to kill the insect pests (you can also use neem oil based pesticides and fungicides). Never use soap based products that say "potassium salts of fatty acids." They work well for garden plants but only add fertilizer to carnivorous plant soil which will also kill them. You can obtain those products at any garden center or plant nursery.

One question, is the pure water distilled or drinking water? Distilled water has all minerals removed but drinking water has minerals added, which will kill carnivorous plants as well.

The rate of yellowing and death points to any number of potential problems like pests, mold or bacterial infection, too many minerals built up, or fertilizer tainting the soil. If you can get more light on your plant it might help as low light is one thing that is a real killer of Venus Flytraps indoors as they simply are not strong enough to resist diseases, pests, and overwintering when dormant. Even my indoors Flytrap under 12,000 lumens of florescent light is slow compared to outside grown ones.

episcesgirl85, are you sure you don't simply have a couple of old traps, dying naturally of old age? The bugs you are seeing in the pot may have nothing to do with the yellowing/blackening -- all kinds of tiny bugs might like hanging out on the wet peat moss -- it doesn't mean they are feeding on the plant.

And traps die all the time -- they each live for, what, maybe a couple of months? And if they are fed, they die even faster, ocasionally even after just one meal, if it's a particularly large bug. And even if not fed, they get old anyway, turn yellow, and start blackening till they turn black completely. That's normal, as long as the plant is putting up new traps -- and if the plant is fed and in good light, new traps will be growing all the time. I guess this trapping/feeding mechanism is complicated, so traps can't last long, so the plant has simply adapted to replace them very fast.

And I have mine indoors too, in a south window -- I wish I could put it outside, but that's not possible for me either (if I were getting a carnivorous plant now, I'd definitely get something more adapted to indoors, but oh well, I'm already stuck with this one). And my plant is doing all right, not pale or floppy at all, nice big red traps, and it even grew 3 offshots this summer.

Anyway -- I guess I think it's an overkill to repot a plant after seeing one black trap. Especially if it's a newly purchased plant, it's probably stressed out as is, I'd let it adjust before doing any repotting. And I'd first find out what the bugs are -- if they aren't harming the plant, why not let them be? Or flood the plant, put it in a big jar with distilled water for like a day -- that would kill most bugs in the soil, wouldn't it? -- And these plants get flooded in their natural habitat, so they shouldn't mind.

Alenka makes an excellent point. If they are just older traps dying, then that's perfectly normal. If it is the new growth that is dying, then that's a concern. Here's a pic of some VFT growing outside (not anymore, they are inside for winter), and I don't know how easy they are to see, but there are at least 6 or 8 blackened traps showing, particularly on the left side of the photo. There are dozens more dead traps hidden under the healthy green traps. This is normal, the old ones die and new ones grow to replace them, as Alenka stated.
-Don
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Alanka does make a good point, however; Flytraps generally do not lose leaves that fast. Mine typically takes a week to lose a leaf enough to make me cut it back just to the petiole. Fast yellowing and blackening usually indicates something systematically harming the plant.

One thing I have observed about plants is that it is best to see how new growth is coming along. If new growth is occurring and all apears well, then it is likely that old leaves are just dying and/or existing leaves have gone through some shock (temperature, humidity,..)but not so drastic as to cause death.