Confusion regarding niches in evolution theory
I think part of the confusion is that these terms are very loosely defined.  Also this discusses evolution in terms of taxanomic radiation, e.g. macroevolution at large spatial and temporal scales.

Problem 1 is that you haven't been clear about what kind of competition, inter or intraspecific competition?  These will have opposing effects generally which can counter balance each other [1]
[2].  When a species is free from interspecific competition, but under high intraspecific competition, that will promote an adaptive radiation, which concurrent morphological changes.  The most famous example of this would be Darwin's finches as studied by the Grants [3].  With a lack of interspecific competition, the finches exhibited rapid morphological divergence.  Following the logic of this, you'll see that lack of competition will lead the kind of rapid taxanomic radiation the video is discussing, especially over a longer time scale.  

However increased interspecific competition will have two effects. It will reduce the number of available niches for an existing species to expand into [4], this generally will drive specialization of a species, which tends to limit genetic diversity which will lead to less "raw material" for evolution to work with. This specialization though is also technically evolution.  However with fewer niches, the new species tend to arise via replacement of existing species in an existing niche, as opposed to an adaptive radiation.  

So you can see how actually both conclusions are right depending on the type of evolution and competition being considered.  

More interspecific competition -> More specialization (one type of evolution), less adaptive radiation (another type of evolution)

More intraspecific competition -> More adapative radiation, less specialization.

I hope that helps answer your question.