Dragons are really good at interpreting sensory feedback from their nose, tongue, and ears.
The 3.5e sourcebook Draconomicon gives a rather precise explanation of a dragon's blindsense ability (p. 18):
One outstanding example of a dragon’s sensory prowess is its blindsense—the ability to “see” things that are invisible or completely obscured. By using its nose and ears, and also by noticing subtle clues such as air currents and vibrations, a dragon can sense everything in its immediate vicinity, even with its eyes closed, when shrouded in magical darkness, or when swathed in impenetrable fog. Of course, some phenomena are entirely visual in nature (such as color), and a dragon that cannot see cannot perceive these phenomena.
Basically, it is just really good at interpreting sensitive feedback. The book gives some more info about each individual sense. In particular, a dragon's sense of smell is exceptional:
A dragon’s sense of smell is nearly as well developed as its vision. This refined sense of smell is only partly dependent on the dragon’s sensitive nose; it also uses its forked tongue to sample the air, just as a snake does. A dragon’s ability to sense the presence of other creatures by scent makes it difficult to catch a dragon unawares, and hiding from a dragon is nearly impossible once a dragon is close enough to pick up the quarry’s scent.
Their hearing is similar to humans, but they excel at interpreting what they hear:
A dragon’s ears are about as sensitive as human ears, and the range of tones a dragon can hear is similar to what a human can hear. Even the youngest of dragons, however, has sharper hearing than a typical human, thanks to its ability to recognize important sounds for what they are and to filter out background noise and focus on significant sounds.
We see this reiterated in the description of the aptly named spell, Hide from Dragons (p. 114):
Dragons cannot see, hear, or smell the warded creatures, even with blindsense. They act as though the warded creatures are not there. Warded creatures could stand before the hungriest of red dragons and not be molested or even noticed. If a warded character touches or attacks a dragon, even with a spell, the spell ends for all recipients.
Moving into 5th Edition, the definition of blindsight linked in the question, which comes from the intro to the Monster Manual, actually answers your question, in not so many words:
Creatures without eyes, such as grimlocks and gray oozes, typically have this special sense, as do creatures with echolocation or heightened senses, such as bats and true dragons.
Bats are the example of a creature with echolocation, and true dragons are the example of a creature having heightened senses.
Unfortunately, this is the extent to which 5th Edition material explains a dragon's blindsight or other senses, as even Fizban's Treasury of Dragons, the most complete volume on dragons published for 5th Edition, seems to omit any exposition on dragon senses, beyond just stating that dragons typically have blindsight. In the "Customizing Dragons" section (p. 34), we see:
Special Senses. Most dragons have blindsight and darkvision. You might upgrade blindsight to truesight, or you could give a dragon with a burrowing speed tremorsense. (See the introduction to the Monster Manual.)