why is the base of my tooth chipped near the gum?

While the toothbrush often gets the credit, it usually is simply "finishing the job" that the bite started.  As the bite forces affect the tooth (or teeth, sometimes several teeth are involved but you only notice one) the flex weakens the enamel.  The weaker enamel is easily brushed away, leading to dentin/root exposure.  Dentin is much softer and now the toothbrush bristles can wear it way much faster, leaving a notch.A filling in this area is a short term solution, but it won't last.  Get the bite evaluated, then decide on a long term strategy.

What you are describing can be a number of things--if it is a smooth wedge-like defect it is most likely what is called an abfraction, some dentists call it cervical corrosion or erosion. This is caused by either toothbrush abrasion or heavy biting forces causing the tooth to flex and break down.Many people have this--it is usually slow  growing and easily monitored.On the outside chance it could be something less common and serious--this is  external resorption--this is less understood and is more erosive and destructive and can eventually lead to tooth loss--the lesions are much more ragged in appearance and advance faster but do not usually involve more than one tooth. See any dental professional and they can easily diagnose the area for you.

It's not likely to be a problem with your wisdom teeth. The tooth chipping at the area you're describing is commonly due to how your teeth come together. If your occlusion is off, or you're constantly clenching/grinding at night then that puts extra force on a tooth causing it to flex and chip. Is also possible you have a small defect or cavity in the tooth.