Why is it so hard to prove a number is transcendental?

For example, $\pi$ and $1-\pi$ are transcendental, but $\pi+(1-\pi)=1$ is not.

It is mainly because transcedental numbers behave so weird. For example, would you think that a transcedental number raised to an irrational is an integer? Well, it is possible:

$$(2^{\sqrt{2}})^{\sqrt{2}} = 2^{(\sqrt{2})^2}=4$$

A difficulty of showing those numbers transcendental lies in the fact that they are not numbers that seem to occur in a natural way.

Sure, they are the sum, product and so on of two of the most common constants, but in "actual mathematics" they rarely appear in this form, and as illustrated in other answers summing and multiplying might not preserve transcendence.

But on the good side $e^{\pi}$ is known transcendental!

(Also $e+\pi$ or $e\pi$ is transcendental as otherwise $\pi$ and $e$ being roots of the polynomial $x^2 - (e + \pi )x + e \pi= (x-e)(x - \pi)$ yields a contradiction to them being transcendent.)