Robbie Burns is a people's poet, aside from the reasons stated above, because he adds a lot of his own personality and spark, flavor and witticism, not to mention the Burns-coined "blue devilism" and malice that comes with being a larger than life person, into his works.
He has no reservations exploring a topic to the core, and his expressions of love, admiration, affection, and sympathy are a mirror of what the common man feels. He does not go overboard trying to accentuate his vocabulary in order to convey intellect, on the contrary, he places higher value in the feeling that in the form.
One of the reasons that Robert Burns was considered a people's poet was that, unlike many of the other poets of his time, he came from the ranks of the ordinary people. So, in some ways he spoke with the people's voice on issues that concerned them, as well as more academic topics. Another reason that the people loved him so much was that he also wrote about these topics using their own much-loved native tongue, making much use of Scottish colloquialisms and dialect. Even readers who were not Scottish loved the musicality this brought to his work. He also put their dear native land on the map, and collected their old airs and songs for no profit - for all of this he is still much loved.
In my opinion, Robert Burns is known as a "people's poet" because of the themes of many of his poems. In many of his poems, he expresses his sympathy for the problems of the common people.
Perhaps the best example of this is not even a poem about people -- "To A Mouse." In this poem, Burns shows that he cares about the weak through his concern for the mouse. By showing concern for a mouse, he is showing that he cares about all beings who are vulnerable.
I think that the fact that Burns tended to write in dialect also shows that he was more of a people's poet than some others.
Additionally, he wrote songs, not just poetry -- perhaps even more of a popular medium.
Finally, I think that the fact that he was born poor helps to make him seem like someone more connected to common people.