1 Prince Rhaegar kidnapped Lyanna Stark
Prince Rhaegar kidnapped Lyanna Stark , the one Robert loved so much. This was the spark that led to the Rebellion.
2 Ned's father and brother failed to stop Rhaegar
Ned's father and brother went to King's landing to ask the Mad king to stop Rhaegar but he ended up killing them both.
3 Robert led the rebellion against the Targaryens
Robert Baratheon led a Rebellion against the Targaryens who were i control of the throne.
4 Robert defeated Rhaegar
During the Rebellion Robert managed to defeat Rhaegar.
5 Jaime Lannister killed the mad King
During the Rebellion Jaime Lannister killed the mad king by stabbing him in the back. (See Why do the Lannisters hate the Starks?)
6 Robert had a better claim for the Throne
After the Rebellion succeeded Robert was the one who had the strongest claim to the throne because his grandfather was a Targaryen.
7 Ned Stark and Jon Arryn didn't interfere
Ned Stark and Jon Arryn didn't want to claim the Throne and so the way was Paved for Robert to become King. (See Why is Bran Stark going north of the wall?)
8 Robert's story was popular
To the common people it was clear that the Rebellion was started by Robert and that it was his right to be the king after it was over.