Why presheaves are generalized objects? While self studying category theory (Yoneda lemma), I came across the statement that for any category $\mathsf{C}$ the functor category $\mathsf{Fun}(\mathsf{C}^{op}, \mathsf{Set})$ represents generalized objects of $\mathsf{C}.$ 
Here generalized means bunch of objects of $\mathsf{C}$ glued together.
Because of the Yoneda embedding $$Y:\mathsf{C}\to\mathsf{Fun}(\mathsf{C}^{op}, \mathsf{Set}),$$ I can imagine that $\mathsf{C}$ lives inside of $\mathsf{Fun}(\mathsf{C}^{op}, \mathsf{Set})$ as $Y(\mathsf{C}),$ however I can not see why other objects in this category acts like generalized objects of $\mathsf{C}.$ 
Can anybody explain me why this philosophy works, possibly with some example.

The previous answers are very good, but I also like to always keep in mind a simple example when working with presheaves, to get a feel for what all this means.
Luckily, we have a very simple and intuitive category of presheaves to wrok with. Consider the category $\mathbb{G}$, whose objects are $[0]$ and $[1]$, and whose non identity morphisms are $\sigma,\tau : [0] \to [1]$. A preseaf $X$ over $\mathbb{G}$ is giben by two sets $X_{[0]}, X_{[1]}$, together with two applications $s,t : X_{[1]}\to X_{[0]}$. You might recognise from this the definition of a graph (or maybe you call it a multigraph, since you can always have multiple egdes between two vertices, but I will call these graphs in the following). Explicitly, $X_{[0]}$ is the set of vertices, $X_{[1]}$ the set of edges, $s$ associates to each edge its source, and $t$ associates to each edge its target.
You can work out that the representable $Y([0])$ is actually the graph consisting of a single point, and $Y([1])$ is the graph consisting of a single arrow between two different points.
Rephrasing your statement for this special cases reads "graphs are a generalisation of points and arrows". I find this quite enlightening, to understand what "glued together" means and how the original statement should be understood
