Applications of $f(x_0)=f'(x_0)$ If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$.
And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.

I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?

I am referring to a case like $f(x)=x^2$ and $f'(x)=2x$, they intersect at $(0,0)$ and $(2,4)$ what is special or can be derived from these points?
If I'm interpreting this correctly, I believe you are asking if there is anything special about this happening locally, i.e at a certain point $x_0$, does the fact that $f(x_0)=f'(x_0)$ mean anything? I would say no; note that by just translating the function appropriately, you can make this happen at any point. To see this, for any differentiable $f$, and any point $x_0$ define $g_{x_0}(x)=f(x)+(f'(x_0)-f(x_0))$, and this function will have that property at the point $x_0$.
Well, $f(x) = f'(x)$, for every $x \implies f(x) = Ce^x$, C is a constant that can be determined if you have a inicial condition. 

There are many applications, such as radioactive decay or growth of some population!