Short answers, pulled from the story.
Annie Besant was elected president of the Indian National Congress in December 1917, making her the first woman to hold that office. She had joined the Congress and, in 1916, co-founded the All India Home Rule League with Lokmanya Tilak to campaign for Indian self-government, modelling the effort on Irish nationalist practices.
Besant and Charles Bradlaugh were prosecuted in 1877 for publishing Fruits of Philosophy, a book by American birth-control campaigner Charles Knowlton. They were found guilty but released pending appeal, and the verdict was ultimately overturned on a technical legal point.
Jiddu Krishnamurti was a South Indian boy, born in 1895, identified in 1909 by Theosophical Society leader Charles Leadbeater as the probable vehicle for a coming World Teacher. Besant became his legal guardian and regarded him as a surrogate son. In 1929 Krishnamurti rejected the role, dissolved the Order of the Star in the East, and left the Theosophical Society, though his personal affection for Besant never waned.
Besant wrote about conditions at the Bryant and May match factory in Bow in her journal The Link after attending a Fabian Society talk on female labour by Clementina Black. The workers, mostly young women, faced very low wages and diseases including Phossy jaw. Historian Louise Raw, in Striking a Light (2011), has contested the traditional claim that Besant led the strike, arguing the primary evidence does not support it.
Besant became president of the Theosophical Society in 1907, after Charles Leadbeater resigned over a controversy and she took over the leadership. By then the Society's international headquarters were located in Adyar, Madras, the city now known as Chennai.
Besant helped establish the Central Hindu School in 1898 and was instrumental in merging it into the Banaras Hindu University, which began functioning on the 1st of October 1917. Neighbourhoods and roads in Chennai, Mumbai, Patna, and Vijayawada bear her name, and the Besant Hill School of Happy Valley in California was named in her honour after her death in 1933.