Why is the Western wall of Jerusalem sacred?

It's because it's a remnant of the wall that surrounded the Jewish Temple, and is located in the old city of Jerusalem, at the foot of the Temple Mount.

(Note: There is no such thing as a wailing wall. This is a misnomer imposed by non-Jewish people who misunderstood the dinstinctive style of Jewish prayer, and thought it was wailing.)

Jews pray at the wall because for many different personal reasons, but in general, it is a place of prayer and hope.

Jews pray there NOT because God is closer to them there, but because they feel God's presense more clearly there.