Bread is mostly flour and water. Flour comprises a small amount of protein (gluten, which is responsible for the bread's elasticity or chewiness) and a large amount of starch (specifically two molecules called amylose and amylopectin).
The process involved in bread baking - in massively oversimplified terms - accomplishes two things:
The latter - starch gelatinization - is the important part in the context of this question. In order for the starch to gelatinize, it needs to be completely dissolved in water and then heated, which breaks up its original crystalline structure. This process cannot be reversed... except, it sort of can.
Gelatinized starch undergoes another process called retrogradation. At lower temperatures, these starch molecules will actually start to realign themselves back into their original crystalline structure or something similar, and during the process they will force out water. That is why refrigerated bread often appears to have a soggy exterior, and frozen bread may develop a layer of frost on the top.
This process doesn't happen on a large scale, but it is enough to make the bread go stale.
But remember that bread is mostly water. When you heat it again, as in the microwave, you are significantly improving the solubility of these reformed crystalline molecules, which causes them to dissolve again. Essentially you are re-hydrating the dehydrated (stale) bread with its own water.
As you've noticed, the taste isn't great. You can't change the fact that the bread has lost water, and a non-trivial amount of great protein and other flavour in the process. The reheated bread is kind of limp and soggy and fragile because the protein (gluten) is what was holding it together. But it's softer.
If you've got very stale bread, another trick you can use is to wrap it in a cloth dampened with hot water for a few minutes, or use a paper towel and microwave the whole thing for a short time. That will do a lot more to hydrate the retrograded starches instead of relying on whatever water is left in the super stale bread.
It sounds like the breadmaker is not cooking properly. If the bread dough feels correct but is the baked loaf is fragile and sticky inside, it is under-cooked. This could be a result of problems with a thermostat, heating element, or possibly the timer.
To solve this problem, try taking the bread maker out of equation by baking in an oven. Use an inverted, oiled sheet pan, and hand-shape the dough (The Fresh Loaf has instructions), or use an appropriately-sized loaf pan. Check the bread 5 minutes before it should be cooked (to make sure it doesn't over-bake due to using oven instead of bread-maker), and keep checking it every 5 or so minutes until done.
If this solves the problem, use the breadmaker for mixing and not baking... or replace/repair it.
Best Answer
Hot gasses such as steam and CO2 trapped inside the dough by the crust are important to help properly bake the bread, as well as to give it form and structure. If you cut a slice off the end of the bread before the bread has finished baking, you completely change the conditions under which the bread finishes baking: steam will escape rather than building up, the internal temperature probably won't rise as much, and the bread will tend to dry out rather than cook.
If you're not sure whether your bread is done, you should take its temperature. An instant-read, digital thermometer with as fine a probe as possible is best because it compromises the crust the least.
For the same reasons, you should let bread cool as @justkt suggests before cutting into the loaf. Bread smells great when it's hot, but it tastes best when it has cooled somewhat.