There are two key reasons from Hutton's era, and several others from the modern era.
The first argument is what we see being formed today. Hutton was the original proponent of the principle of uniformitarianism (a.k.a. "the present is the key to the past").
We can find environments producing similar sediments as those rocks today, and they form slowly. This is the key parts of Hutton's original argument: That cliff is made of sandstones that are similar to those forming on beaches and rivers just around the corner. However, the modern processes we see forming sandstones operate very slowly. We do see some rapid pulses of deposition, but then very long times between each pulse. It takes even longer for the sand to start hardening into rock (with the exception of calcite cementation, anyway, which can happen in a few decades). If you look at the rocks in that outcrop, you'll see that they are themselves made up of other rocks. Furthermore, they have pebbles and boulders of themselves (solidified as rock and then eroded) contained within them. To have a sandstone that has pieces of itself that have hardened into rock, been eroded away, and then re-deposited requires a long time with any of the processes we see operating today.
The next reason is scale. There are lots of processes that can locally deform things (e.g. a landslide), but very few that can consistently deform things in the same way across an entire continent. You can trace that same interval of upturned (folded) rock all the way across the UK (and North America and other parts of Europe, though that wasn't known in Hutton's day). What process could affect such a huge area in a very short time? If it did occur in a short time, why isn't the rock shattered instead of folded? (Materials deform very differently at different strain rates.) How do you fold kilometer-thick sections of rock over thousands of kilometers without it being a gradual process?
There's tons of other very clear reasons, but they weren't known in Hutton's day. The simplest answer is still to go look at the rocks. They demonstrate a clear record of environments you can see today. Coal mines played a large role in convincing people in the 1800's, as they preserve an immediately recognizable environment. You can see coal seams with stumps of trees growing through them (complete with roots). The stumps are snapped off by a river channel that cuts through. Next another coal seam with stumps in it, and then the same above. Do we have any reason to believe that trees would have grown dramatically faster in the past?