What is the purpose of trimming the bottom corners of TLC plates?
Yes, your question contains the answer. Think about capillarity.

Without the little cuts the eluant would be "sucked" and flow from both the bottom and the left ( and right) edges of the thin plate, resulting in a tilted flow that


tends to move the elute faster along the vertical edges while clumping the spots to the center; 
decrease the resolution at the given conditions by deforming and enlarging the spots.


Think of it as a weird bad executed two dimensionally-developed plate, if it helps.

Note that beside resolution, the cuts should be there to guarantee a levelled horizontal and straight flow front, as requisite to obtain reliable and useful ratio front values.

(Good practice: draw the real appearance of the developed plate or even post it to the lab journal, so that a future user get a feeling of the separation whatever it happened. The latter not recommended if a product spots out to be irritant/allergenic).



Left: correct - Right: not horizontal flow frontline due to capillarity at vertical edges. 

As OP mentions an example with cutted corners but not levelled flow, obviously cutting the corners but plunging the plate in too much eluant is useless. The idea is to have eluant sucked up just by the bottom edge.