A literal response to your title question would simply be "because in the physical world, oscillations behave in ways consistent with these functions." Of course, one then wonders why $\sin$ and $\cos$ are so ubiquitous.

Depending on your level of physics background, you may be familiar with simple harmonic motion - that is, motion for which there exists a restoring force proportional to displacement. For example, motion of a spring is simple harmonic (since by Hooke's Law, the restoring force is proportional to the amount a string is stretched) and motion of a pendulum for small angle amplitudes is simple harmonic. Simple harmonic motion, since it's such a simple model, appears quite a bit in the physical world. (More examples are on Wikipedia.)

Quantitatively speaking, we mean to say that for simple harmonic motion, $F = - k x$ for some displacement $x$. Moreover, $F = ma = m \frac{d^2x}{dt^2}$, so combining these two equations, we find that

$$\frac{d^2x}{dt^2} = -\frac{k}{m} x$$

This is a differential equation that must be solved to find $x(t)$. It turns out that the solution to this equation is an expression of the form $A \sin( \omega t - \phi)$ for constants $A$, $\omega$, and $\phi$ - to verify this yourself, plug in a function like $x(t) = 2 sin\left(\sqrt{\frac{k}{m}}t - \frac{\pi}{2}\right)$.

Since simple harmonic motion is the most common form of oscillation, and simple harmonic motion is described using $\sin$ and $\cos$, most oscillations in physics follow these trigonometric functions.

A literal response to your title question would simply be "because in the physical world, oscillations behave in ways consistent with $\sin$ and $\cos$." Of course, one then wonders why these functions are so ubiquitous.

Depending on your level of physics background, you may be familiar with the harmonic oscillator - that is, a system for which there exists a restoring force proportional to displacement. For example, motion of a spring is simple harmonic (since by Hooke's Law, the restoring force is proportional to the amount a string is stretched) and motion of a pendulum for small angle amplitudes is simple harmonic. As a matter of fact, any object in stable equilibrium will move harmonically for small perturbations.

Quantitatively speaking, we mean to say that for simple harmonic motion, $F = - k x$ for some displacement $x$. Moreover, $F = ma = m \frac{d^2x}{dt^2}$, so combining these two equations, we find that

$$\frac{d^2x}{dt^2} = -\frac{k}{m} x$$

This is a differential equation that must be solved to find $x(t)$. It turns out that the solution to this equation is an expression of the form $A \sin( \omega t - \phi)$ for constants $A$, $\omega$, and $\phi$ - to verify this yourself, plug in a function like $x(t) = 2 sin\left(\sqrt{\frac{k}{m}}t - \frac{\pi}{2}\right)$.

Since simple harmonic motion is the most common form of oscillation, and simple harmonic motion is described using $\sin$ and $\cos$, most oscillations in physics follow these trigonometric functions.

The cycloid doesn't appear as much as $\sin$ and $\cos$ simply because there's no reason for it to. There aren't many physical phenomena which follow cycloid paths, since the cycloid is such a complex shape compared to the fairly simple $\sin(\theta) = \operatorname{Im}({e^{i \theta}})$ and $\cos(\theta) = \operatorname{Re}({e^{i \theta}})$.