As noted, there are several types of "fiber" in vegetable foods. Their composition is rather variable, depending on the vegetable.
The following is a very short and simplified overview of the most important factors.
The hard visible fibers like you see in celery are basically cellulose, and won't degrade or soften much (if at all) during cooking (e.g. the outer part of broccoli stems). Most vegetables don't contain much cellulose (there is always some), it's mostly present in the surface layers (which have to protect the interior) and in the visible fibers.
Kale is a bit special, in that it has a particularly thick surface, which also contains some other insoluble (wax-like) components. As neither the cellulose nor the waxes are degraded during cooking, the kale will stay rather firm (the links between surface cells will be weakened a bit, as described below).
Then, there are different processes that do soften vegetables.
In potatoes, an important part of the softening is caused by a gelatinization of the starch, which swells and disrupts the cell walls. In addition, the pectins in the cell walls are dissolved and degraded as decribed in the next paragraph.
In most other vegetables, the pectins in the cell walls partially dissolve and can be broken down. This makes the links between individual cells weaker and allows them to separate (as in carrots and such). In addition, the cells become easier to break and liberate some of the water/juice in the cells (very noticable in e.g. spinach and lettuce). Al of this makes the vegetable softer.
There are some other soluble fibers as well, but they don't play much of a role in the softening process.