It does indeed seem counterintuitive that $L_4$ and $L_5$ would be at the same time both high points of potential as well as stable points in the system. In fact, a quick look at an example contour plot with all five Lagrangian points demonstrated would also suggest that $L_4$ and $L_5$ would be unstable:
In the picture you can see that $L_1 - L_3$ are placed in saddle nodes, and intuitively one concludes that the smallest perturbation would set them "falling" in either direction, i.e. they need station keeping.
However, $L_4$ and $L_5$ would still seem to be unstable, so what gives? What makes $L_4$ and $L_5$ stable is that each of them is located equidistant from both of the masses. This leads to the gravitational forces from each of the bodies towards $L_4$ and $L_5$ to be in the same ratio as the two bodies' masses, hence the resultant acceleration points towards the barycentre of the system (which is also the centre of rotation for a three-body system).
This resultant acceleration, and by extension force, leads to the object in either $L_4$ or $L_5$ to in fact have just the correct amount of acceleration / correct force towards the system's barycentre so that it will not fall out of orbit (it "falls" towards the barycentre as the system rotates, like Earth similarly falls towards Sun throughout its orbit), as counterintuitive as it sounds at first. (Furthermore, this effect is not only limited to bodies of negligible mass, but is independent of mass, which is why you can have more massive bodies orbiting in $L_4$ and $L_5$ like Jupiter's trojans, as well as smaller, artificial satellites.)
Source: Lagrangian point - Wikipedia
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