Why is he called Babasaheb?
Babasaheb is an honorific, not a name, and it tells you how he is held rather than who he was. It joins baba, an affectionate word for father, to saheb, a term of respect — so it means something close to respected father. His followers gave it to him, and it stuck because it fit: to the communities he led he was not merely a leader or a lawyer but a father figure, the man who had gone first and made a way where there had been none.
The site otherwise calls him Ambedkar, following the ordinary convention of using the surname, and reserves Babasaheb for exactly this: the places where the subject is his standing in the hearts of the people he worked for. The name is a piece of evidence about that standing. Statues of him across India, hand raised holding a copy of the Constitution, are addressed by that word, and the anniversaries that fill the roads are for Babasaheb, not Dr. Ambedkar.
There is a lesson in the affection that the political arguments can obscure. A man remembered by outsiders for a single controversy is remembered by millions as family — which is a truer measure of what he was to them than any office he held or any title his degrees earned him.