Why do people remove excess solution after they put on a coverslip?
Neither of the current answers fully answer the question. 

The amount of liquid underneath a coverslip should be just sufficient to hold the coverslip and sample onto the slide. If there is too much water, the surface tension would become reduced, allowing the coverslip to fall off the slide. 

Additionally, minimising the amount of water between the coverslip and the slide would ensure that the z-axis variation in your sample is minimised, therefore reducing the likelihood that your sample will have different focal points across the sample. 

This UCLA guide on microscopy slide mounting explains it (emphasis mine):


  Small pieces of filter paper (Whatman #1) are placed around the edge of the coverslip to absorb excess mounting medium. If your specimen is tissue culture cells or frozen tissue sections, this should not harm them. This does two things: a) it allows the coverslip to come as close to the specimen as possible and reduces the depth of the field. You will get less focus aberration if you do this judiciously. b) the coverslip will be a lot less likely to float around on the slide and move which could specimen damage. The surface tension between the coverslip and slide will be much higher. You will also be much better able to seal your coverslip to the slide.