I found the same image used by ESL, but in English. See below:
In such diagrams, a 4D (3 spatial dimensions + time) universe is "compressed" into two dimensions. Any lines over such a diagram will look like straight lines to observers within that universe. In fact, those lines are paths of light rays.
So if you want to picture what a wormhole would look like, simply draw lines over the diagram that connect an observer to visible objects while passing through the wormhole. The wormhole would seem like a very powerful fisheye lens with whatever is around it seeming to be inside it. There is an artistic rendition in Wikipedia.
As for how it would feel, going down through one. You would be utterly f... Destroyed.
Going through a wormhole would be like traveling over a line that goes through the wormhole. The universe is very curved but it would seem like a straight line to the traveller. The real problem is gravity.
Wormholes have mass. You may imagine them as just a topological feature of the universe, but that topological feature requires exotic matter to be stable. So the mass of the wormhole is the energy-mass of that exotic matter. We could talk a whole about how that energy mass is perceived, but it suffices to say that even if the exotic matter in use would the the kind to produce "negative mass", there would still be forces pulling it towards regular, positive mass. For all practical purposes, the energy mass would be positive.
So how massive would a wormhole feel? If it is one-meter wide and very flat, it will have the approximate equivalent mass of Jupiter, or close to 318 Earths. In another post I describe what would happen should the Earth ever come close to something like that.
Should a spaceship come close to that, it would face dire tidal forces. It would be akin to approaching a black hole. You would be broken apart as you fell into the throat. The difference is that instead of going into a region of space with practically infinite gravity, your particles would be spread out with great force somewhere else in the universe.
If you handwave the monstrous gravity and manage to survive, though, it should feel like the ultimate roller coaster.
Last but not least, as for how long it would take to cross the wormhole... I'll refer to this post in Physics:
There's a recent popularization of wormhole physics that nicely lists the properties of the four wormhole examples that Morris and Thorne considered in their 1988 paper. These properties include traversal times. Here's a summary
(exotic matter distributed throughout space) ~ 1 hour
(exotic matter confined to large finite radius) >= 7 days
(exotic matter loosely restricted to throat) ~ 200 days
(exotic matter closely restricted to throat) >= 0.7 seconds
These times are completely independent of the distances between the mouths of each wormhole in normal space
Suggested theme song for wormhole traversal: Shooting Stars, by the Bag Raiders.