I'm not sure where you got that quote, but it is very vague and not how it should be worded for the general case.
Buckling occurs due to eccentric loads, primarily on slender columns. Slenderness is the ratio of length to radius for a round column. A thin ruler is much more slender than a pop can, because a pop can is less length (shorter) and has a greater effective radius (ruler isn't as wide on any side).
That said: They may be talking about a situation closer to Lateral-Torsional Buckling similar to that pictured on the page:
If you were to increase the depth of the beam, it would be more prone to buckling in the middle in the fashion described there. This is quite a bit different from regular buckling though.
In this case, increasing the depth would be the same as increasing the slenderness ratio for the beam in the vertical direction. A big factor with this type of bucking has to do with the length of the beam though. When you apply the force far away from the supports of a beam it has a tendency to twist and bend (especially if it is more slender, the deeper it is, the bigger the moment you create around the middle when loading the top).
As far as your second question goes, real bridges have distributed loads. If you have to point load the bridge, do it near a support, preferably a very strong one.
If you're trying to build a bridge (sounds like it might be a project), consider that buckling wont be the only thing that can go wrong. Making the beams "deeper" will reduce the bending, but given some circumstances they will still easily buckle through the lateral-torsion.
Here is a (very bad MS paint) diagram I made of the forces.
Obviously in reality the angle would be a lot smaller, but you see if there's and eccentricity of the load, the deeper beam will have a bigger moment, because the force is the same, but $R \gt r$ so the moment on the deeper beam is greater, causing more twisting.