How does time translational invariance and linearity imply exponential solutions?

I’m currently studying "Waves and Oscillation". While going through the book The Physics of waves, from page 11-12. The author has mentioned that the differential equation being linear implies that we can write reduce the solution to $z(t+a) = h(a)z(t)$ and then he directly compares this hypothesis with the $cos$ and $sin$ solution of simple harmonic motion.

My problem is that I don’t get, how can we infer this result form the linearity and the time transnational in-variance?
Also, how can we directly say that the same property as periodic function (like $sine$ or $cosine$) will be shown by our hypothesized solution (i.e. it’ll change sign for some particular value of a)?

On page 19 he has mentioned that "So long as the system has time translation invariance and linearity, the solutions will be
sums of irreducible exponential solutions." What on earth it means?

Thanks for help!