I find the most intuitive way to think about Lagrange points is to consider the gravitational potential energy and a surface representing the potential.
The Lagrange points exist in the space around two massive objects, say the Sun and the Earth. You can think of both of these as producing a gravitational potential well shaped a bit like a cone or funnel pointing down but with sides that make a curved profile. The gravitational potential as a function of distance $r$ is given by $U = -Gm/r$. He's a nice picture grabbed from Wikipedea 2. Image source: By <PERSON> - Own work, CC BY-SA 3.0, Graviational Potential Well
Now, since there are two massive bodies there are two of these wells near each though the one centred on the more massive body will be bigger. You can imagine that there would be a saddle point between the two wells were the surface would be flat. This would represent a point where the forces of gravity pulling towards each body are in balance. (I couldn't find a good image of two wells.)
The thing is that the smaller body (say the Earth) is orbiting the larger body (Sun) or more accurately they are both orbiting their common barycentre. The Lagrange points are stationary as seen from the Earth so we need to consider things from a frame of reference that is rotating with the Earth. 921
In such a rotating frame of reference we have to introduce a fictitious centrifugal force away from the centre of rotation. This the addition of this force changes potential surface with the gravitational wells. Since the force is acting away from the centre an object near the centre would have a higher potential than one farther away so the potential for this force would look kind of like an inverted bowl. When you put together the effects of gravity from both bodies and the centrifugal force you get somewhat complicated effective potential surface. Here's an image, thanks once again to Wikipedia.
Image source: By User:<PERSON> - Own work, CC BY-SA 3.0,, Lagranian Points Equipotential
So, the five Lagrange points are places where the slope of this potential surface is zero. You could imagine yourself sitting there and not sliding away from them if you were careful. The $L_1$, $L_2$, $L_3$, are all saddle shaped whereas the $L_4$ and $L_5$ points occur at local maxima. They are like hilltops but quite flat hilltops.
Nonetheless, the the $L_4$ and $L_5$ points are not local minima so how are they stable? Well, in the rotating reference frame the centrifugal force is not the only fictitious force that has to be introduced. Once something starts moving it also experiences a Coriolis force perpendicular to its motion. If a satellite moves away from the $L_4$ or $L_5$ points it's this Coriolis force that will nudge it back if not exactly back to the Lagrange point then to an orbit around it. 246