The problem with understanding anything bigger than 3D is that we live and think in 3 dimensions. Going to 4, 5, 6, ... dimensions is similar to trying to draw a 3D picture on a piece of paper. You can get the idea of a 3D object across by shading and perspective tricks, but in the end, you're really just drawing a 2D object because you're working on a 2D piece of paper. It's tricky to think about 4 dimensions because--well how would you represent a 4 dimensional object by using familiar 3 dimensional ideas?

However, if you take a step back and ask "what is the mathematical meaning of a dimension" you can go ahead and describe higher dimensions mathematically: It's basically just the number of coordinates you need to specify in order to describe an object. I think an easy way to get your head around the idea is to think in terms of what use you might find for 4+ dimensions in real life. How many dimensions do you need to specify:

How far the car has gone? 1 dimension for distance.

Where the car is on a map? 2 dimensions, one for latitude and one for longitude.

Where the car is on a topological map (i.e. where is it on a map and what is its altitude?) 3 dimensions, latitude, longitude and altitude.

Where the car is on a topological map and what its speed is? 4 dimensions, latitude, longitude, altitude, and speed.

Where the car is on 2D map, and what it's speed/direction is on that map? 4 dimensions, latitude, longitude, north/south speed and east/west speed.

Where the car is on a 3D map and its total speed and direction? 6 dimensions, 3 for speed/direction + 3 for position/altitude.

There's other uses for 4+ dimensions in physics, depending on what you're trying to plot. A really famous one is the 4 dimensions you get in relativity "space-time" which is 3 dimensions (space) + 1 dimension (time). The physical meaning of the higher dimension depends on why you're using them. In relativity, its thought that the fabric of the universe itself is 4 dimensional. In my examples above, the 4 dimensions are a mathematical tool for describing the car.

You can extend shapes to any dimension by extending their definition to higher dimensions. For example, a sphere is "all points that are equally distant from a central point." As long as you know how to measure distance in your space, you can plot a sphere. Usually (but not always) distance is given by the "Pythagorean theorem" which says that the distance between two points can be calculated by adding the squares of the distance between them on each axis, and then taking the square root of that number.