There is no explanation in Roman literature itself of the purpose of Hadrian's Wall.

Historians have different theories about the purpose of the wall, from being able to impose customs duties, and control movement, to simply marking the border. The problem with the customs-duty theory of the wall is that the people in the north were too poor to make any income derived from it worth the cost and trouble of building it.

Perhaps the best theory is that its main purpose was to prevent small-scale cattle raids across the border.

Without the wall, twenty or thirty men could cross quickly into Roman territory undetected, raid and loot an isolated farmhouse, make off with the cattle, maybe even take some women, cross rapidly back over the border, and disappear into the Pictish hinterlands.

That kind of raid can be very difficult to prevent. It might be some time before the military was even notified. Following the raiders over the border and finding them would require a major military expedition, with little chance of success.

However, with the wall, small raids like that are effectively eliminated. It's far more difficult to cross without being detected, both coming and going, and most importantly, cattle (which would be the main object of raids) can't be moved across it.

So the wall would give a lot more security to farmers near the border.