Imagine you're looking at a mountain peak that looks like an eagle's head. This eagle head "does not exist from its own side" - it is you, the observer, that delineates the head from the rest of the mountain range and imputes the features such as the eyes, the feathers, the beak etc. to construct the image of an eagle. That imputation or projection from observer's memory onto external stimuli and resulting delineation, designation, and recognition of a subset of that stimuli as a certain known entity - is what's simply referred to as "a mere conceptual construct".
The same process is behind recognizing images in the morphing clouds, on the starry night sky, in the tree canopies.
The same process is behind production of dreams from the random noise our sense organ circuitry generates during sleep.
It takes only a little more effort to understand that the same exact process is also behind a one year old infant parsing it's sensory input into distinct entities such as "mother", "dog" and "ball". None of these exist as separate distinct immutable entities over long time, but are delineated and designated by the infant brain based on the sensory impressions accumulated during the first year of life.
Even us, adults, live in the subjective world of objects that don't exist from their own sides but are conceptual constructs we were taught to delineate and designate.
Hope this helps.
See this inspired answer for better examples and a clearer connection to the concept of Emptiness.