From the Wikipedia page , which has more detail
The Crystal Palace and South London Junction Railway was authorised to build a line from Peckham Rye Station to a terminus at Crystal Palace in 1862, in order to serve the attraction of the Crystal Palace.
The station was opened by the London, Chatham and Dover Railway on 1 August 1884. It was temporarily closed from 1 January 1917 until 1 March 1919, and again between 22 May 1944 and 4 March 1946. Permanent closure occurred on 20 September 1954.
Upper Sydenham station was the penultimate station on the route before Crystal Palace, sitting in the wooded area at the top of Sydenham Hill, overlooking Dulwich and Sydenham. The station and the line was poorly used despite new houses being built in the area as passengers preferred to use other stations near-by (Sydenham Hill Station, Crystal Palace (Lower Level) Station and Sydenham Station ) which were on more direct routes. The line was one of the first of the former South Eastern and Chatham Railway to be electrified by Southern Railway, under "South Eastern Electrification - Stage 1" in July 1925. The destruction of the Crystal Palace in 1936 saw patronage reduced.
During the second world war the branch line suffered damage from enemy action. After the war the cost of repairing the war damage with declining receipts led to the decision to close the line to close on the 20 September 1954, the first permanent closure of an Southern Electric line. The dismantling of the station took place in 1957.