You're correct that clocks, as physical devices, don't have any inherent awareness of time dilation or the fact that they are measuring time. However, the concept of time dilation is based on the behavior of physical phenomena, including the behavior of clocks, as observed by different observers in relative motion.
When we talk about the effects of time dilation, we are comparing the measurements of time made by different observers in different reference frames. In the scenario you described, where one clock remains on Earth and another travels at speeds approaching the speed of light, the observers associated with each clock would perceive time differently.
From the perspective of the observer on Earth, who is relatively stationary, the clock on the spacecraft traveling at nearly the speed of light would appear to be running slower compared to their own clock. This means that fewer seconds would elapse on the moving clock for every second that passes on the Earth-bound clock. The observer on Earth would see the moving clock lagging behind.
Similarly, from the perspective of the observer on the spacecraft, who is also moving at a high velocity, their own clock would appear to be running normally. However, they would observe the clock on Earth to be running faster compared to their own clock.
The key point to understand is that when two observers are in relative motion, their measurement of time will differ due to the effects of time dilation caused by their differing frames of reference. The observed time dilation arises from the fundamental nature of spacetime, as described by Einstein's theory of relativity.
While clocks themselves don't "know" that they are measuring time or experience time dilation, the measurements they provide can be compared and analyzed by observers in different frames of reference to observe and quantify the effects of time dilation.