Off-road tires with big lugs and deep grooves are designed to sink in to the soil and "push" against the soil (shear forces) in addition to friction with the ground. In contrast to slick road tires that are relying almost entirely on friction between the tire and ground. Treads in slick and off-road tires help with traction in the wet by evacuation water from contact area of the tire by simply giving the water a place to flow.
W/ respect to contact patch: friction does not depend on surface area (this is true), but when you are riding off-road, the wider contact patch (larger contact area) is better because you are applying forces to a larger area and are less likely to "deform" the soil under the tire (slip). When you are riding on-road, asphalt is so much stronger than loose soil that you don't need to worry about it deforming under your tires, so we can have nice, efficient skinny tires (until you run into some sand or gravel on the road)
Hope that clarifies a little bit
