The Wailing Wall was not part of the Temple - it was part of the Temple Mount - and a giant retaining wall for the courtyard on which the Temple sat. Here is a model of what we think it would have all looked like:
Basically, when Herod restored the Second Temple, he couldn't expand the building itself, since it's dimensions were fixed by Scripture. He wanted something grand, however, so he built a giant retaining wall around the top of Mount Moriah. The Temple itself sat atop the mount - the courtyard stuck out from the hilltop.
When the Romans destroyed the Temple in 70 AD, the building was levelled, but the mount itself remained. What you see today is the Dome of the Rock (would be to the right of the Temple in that picture) and the El-Aska Mosque to the Left. The Courtyard remains.
When you visit today (and I have!), you are at the base of the retaining wall, on the side you can't see from the reconstruction. (This model is at the Holyland hotel, and well worth the visit itself!)
Finally, you will note the "wall" perpendicular to the mount. That is the remains of Wilson's Arch the road that would take the worshipper up to the Temple. It seems to empty out right between the current structures.