Manhattan distance is ∣x1−x2∣+∣y1−y2∣. It measures distance in a grid where you can only move horizontally or vertically. This is different from Euclidean distance (straight-line distance). Manhattan distance is easier to compute: no square roots needed.
For MST purposes, Manhattan distance works like any other edge weight. Compute it and treat it as a cost. The algorithm does not care about the formula.