The easiest way to go from 3D to 4D, is to first reflect on how we go from 2D to 3D. So for that, lets first go to 2D, so we can come back to 3D.
Imagine a very large flat sheet of glass with a very small thickness. Then imagine someone snapping a collar onto your neck, fixing it at ninety degrees from your spine. Now you can only turn your head left and right, but not up or down. The key to the collar is put in your pocket. Then suddenly you are miniaturized into a speck just about as high as the thickness of the sheet of glass, and inserted into the sheet from a “door” in one of the edges.
This sheet of glass is your world now, the door was only one way. You explore your world, moving in random directions left and right, meeting other folks like you. Soon you come to a wall. You check it out, but it seems to go really far in both directions. The only way around it, your new friends tell you, is to walk until you reach its edge. Uh..”Why can’t I just climb it and jump over?” you ask. Weird stares greet you. “Is this guy from another dimension!!?”, someone quips, while scratching what looks like a wart on the back of his neck.
Nevertheless, you remove the key from your pocket, and press it onto the wart on your neck. “Hey where did you get that key? Will it work on my neck-tie too?”. You turn around and smile at them, and then, to everyone’s horror, you bend your neck and look UP. You run toward the wall as fast as you can, and at the last moment, jump. A moment later you find yourself on top of the wall, looking down at your friends, from outside the sheet of glass. “Hey where did he go?”,”He disappeared into thin air!!”. You turn around, jump off onto the other side, to the amazement of a pair who see you materialize from nowhere.
You can now traverse the third dimension! Whenever you encounter an obstacle too big to be circumnavigated, you simply jump over it in the third dimension. You are now an enlightened being in this world. To your friends, you have the ability to walk through walls!
In a 2D world, there was a 1D obstacle, which you circumvented with ease by jumping over it in the third dimension.
Now imagine that you are back outside the sheet-of-glass, normal as the rest of us in our 3D world. A few days later, while hiking in an unknown part of the woods, you come across a wall. A really large wall. It goes as far as you can see in both directions, and surprise, it goes as far high as you can see.
You stand there, dumbfounded. Soon enough an acquaintance comes by. He looks around, just like you did, but then he does something unexpected. He smiles, and takes out a strange looking key from his pocket. Inserts it into his belly button, and turns. Then, to your horror, he cool-ly bends his back backward in such a twisted manner that his forehead touches the back of his knees. In this contorted position, he walks back a distance from the wall, and starts running toward it. At the very last moment, he disappears!
As you begin to comprehend that he circumvented the 2D wall by jumping over it in the fourth dimension, you pick up the key he left behind.
But alas, it feels too small for your belly button.