What do the Jews believe the western wall to have been a part of?

Answer 1

The Western Wall is part of the Temple Mount of the Second Temple. This fact (not "belief") is recognized by historians as well.

Answer 2

Jerusalem is a hilly city, so in order to have a complex as large as the Temple Mount, you needed to create a level esplanade. The northern and eastern sides of the area were at a higher elevation than the southern and western sides. As a result, retaining walls were built on the southern and western sides to create a level esplanade. The Western Wall is the western retaining wall of the esplanade. The reason why this is holy is because the Jewish Holy Temple was situated just above the central part of the Western Wall before the Romans demolished it.