In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as shadows of 4-D objects. How does this helps us visualize 4-D objects?  

 I searched that we can atleast see their 3-D cross-sections. A tesseract hypercube would be a good example. Can we conclude that a 3-D cube is a shadow of a tesseract?  

But I can't understand how can a shadow be 3-D? was the screen used for casting shadow also 3-D; or else what way is it different from basic physics of shadows we learnt? (I want mathematical viewpoint for this question, hence asking its physics here) 



Visualizing the 4th dimension
From a mathematical viewpoint, the dimension is just a number. You can of course define something like a "shadow" as the intersection of an object in 4-D and a subspace of dimension three, but this is but a definition. It might help to not think too much about how it would look like (because, as you noticed, you can't really think in four or more dimensions) but only take the definition and work with it.
If you have a proper definition of shadow, e.g. 
$$S : \mathbb{R}^{n + 1} \to \mathbb{R}^n,$$
you might want to check that for $n = 1$ and $n=2$, where we can in fact imagine things, $S$ coincides with the natural concept of a shadow. This should be enough to call $S$ a shadow in every dimension.


