If we ask what is time that we experience, the question might be slightly more of a philosophical nature. However, it is an observational fact that we experience time in the sense that we can clearly tell if two events are occurring simultaneously or not. For example, if we hold two flash lights in our hands and observe when they flash, we can tell afterwards if they flashed simultaneously or one after another (assuming there is no biological limit of ~0.1 second on our eyes to distinguish between visual inputs).
Now if we want to quantify this notion of non-simultaneity or want to compare different instances of non-simultaneous events, we will need a scale to measure it. In this case, we call this entity to be measured as time interval. The scale can be a collection of non-simultaneous events and we can define the unit of time to be the time interval between two subsequent scale events. For example, we can drop a tennis ball and define the bounces of the ball off the ground to be the scale events. Then, the time interval between two bounces will be a unit of time, say we call this unit as 'bouncegap'. Now, if we drop the tennis ball in such a way that the first bounce coincides with when the first light flashes and the second light flashes at the third bounce, then we can say the time interval between the flashes is 2 bouncegaps. Now, this definition of unit of time is not universal, therefore not preferable as a standard, and so are some units of time like lunar month, solar year, solar day etc. We define second to be a unit of time that can be measured to be the same amount of time interval by any observer and the definition given by BIPM is:
The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of 0 K.
In this definition, the cesium atom is not oscillating to have a frequency, rather it's at rest. When an electron in this atom transits from higher to lower hyperfine level of the ground state, it emits a photon with energy $3.802\times10^{-5}$ eV and frequency 9192631770 Hz. Therefore, every observer in the rest frame of the cesium atom will measure the duration of 9192631770 periods of this photon to be 1 second.
To answer your second question, a cesium atom is also able to experience time. Suppose, the cesium atom emitted a photon due to a transition between the ground state hyperfine levels. Now for example, a second photon can be emitted after one time period of the first photon due to any other electron transition in this atom. Then the time experienced by this cesium atom between these two emissions would be 1/9192631770 second as shown in the following spacetime diagram.
The time dilation in special relativity arises from the invariance of the speed of light $c$ in different inertial frames. One of the easiest ways to demonstrate the emergence of time dilation from invariance of $c$ is to use the following thought experiment. Suppose, an observer has a clock to measure time in his/her rest frame. But instead of measuring the frequency of a radiation from cesium, s/he has a setup of two mirrors (with 100% reflectivity) at a distance $d$ apart and a photon is set to oscillate in between those two mirrors indefinitely. This observer will measure the time duration of one oscillation of this photon in between the mirrors to be $\Delta t = \frac{2d}{c}$. If $d$ were 149896229 m then $\Delta t$ would be 1 second and this 1 second would be the same 1 second if s/he would have measured it with a cesium atomic clock.
Now, suppose there is another observer who has an identical photon clock and s/he is moving with a velocity $v$ relative to the first observer in a direction perpendicular to the clock length. This observer would see the photon oscillation in his/her own clock to happen just as the first observer sees in his/her own as shown in figure (a). However, if the first observer looks at the clock of the second moving observer, s/he would see the photon oscillation to be like figure (b). This is because the mirrors in the photon clock are moving and when a photon reaches a mirror, the whole clock is at a different point in space than where it was when the photon left the other mirror as seen by the first observer. Now, this first observer will think that the second observer is actually measuring the time period of the photon clock to be $$ \Delta t' = \frac{2d}{(c^2 - v^2)^{1/2}} > \frac{2d}{c} = \Delta t.$$ Thus, the first observer will conclude that the second observer's clock is moving slower than his/her.
