May I ask how a tripod can be unstable?
Unstable might be the wrong word to describe the major problem with tripods; unsteady might be a better choice. Tripods are quite stable in the sense that, absent a really strong wind or careless photographer, all three feet will generally stay on the ground. They don't rock like a four-legged table on uneven ground.
However, all tripods are unsteady to some degree. Much of that unsteadiness comes from the legs, which are generally made of several sections of some sort of tubing that telescope inside one another. If you looked at just one section, even the smallest one, it'd seem very rigid on its own. When the leg is extended to four or five feet, though, there can be enough flex in the whole leg to be a problem.
How much flex are we talking about? If you set the tripod up with the legs extended all the way, it's not uncommon for the legs on a cheap tripod to deflect a centimeter or more if you apply some force near the middle of the leg's length.
The three legs are joined at the top to a bracket called a spider, and those joints are another source of shake. On some (usually more expensive) tripods those joints lock in position; on others, they don't lock at all, but cross braces limit the distance the legs can spread.
Tripods typically also have a center column that extends through the spider. Since the tripod head, and ultimately the camera, are mounted to the top of the center column, any play in the connection between the column and the spider creates some potential for movement.
One way that photographers try to eliminate shake and dampen vibration is to preload the tripod with some weight. Good tripods often have a hook on the lower end of the center column for just that purpose -- you can hang your camera bag, a bucket of sand, or any heavy thing you have handy from that hook, and that adds a lot of stability to the tripod. Of course, that can backfire, too -- if a weight suspended from one end of the center column starts swinging, the camera at the other end is going to see some movement. Rock bags are less prone to this problem because they attach at three points.
Finally, the head is one more source for unsteadiness. Cheap tripods generally come with cheap heads, and cheap heads are less likely to be as rigid or to lock down as tightly as better heads do.
Also, please apply the answer to my situation: they'll all be landscape photos where I have all the time in the world (so I'll be using a remote shutter/2 sec timer), and I will be shooting long exposures (5 to 20 secs). Plus the place I will be going will not be that windy.
The reason that people get worked up about the stability of a tripod is that even very small movements have a noticeable impact when using a long lens and/or making a long exposure. People spend a lot of money to buy very sharp lenses, so it doesn't make much sense to then use a cheap tripod that'll introduce a lot of motion blur. Even the tiny force of a DSLR's mirror flipping out of the way can create an unwanted vibration for an astrophotographer.
If you're shooting landscapes with a wide angle lens, for example, you might not be quite as concerned about absolute steadiness. More steady is always better, but steady enough for your purposes may well be attainable without spending a big pile of dough.