Will a high gain antenna help transmit through obstacles if the radio path is non-line-of-sight (non-LOS)?
First, please understand exactly what RF Line-of-Sight (LOS) means.
Secondly, Digi recommends using only medium gain antennas in non-LOS conditions. The very high gain antennas actually perform worse in non-LOS conditions because they have a narrower beam width. The multipath effects of a non-LOS environment cause RF signals arrive at the antenna at odd angles after reflecting off of nearby objects. A lower gain antenna with a larger beam width will be better at recombining the bounced signals.
Generally speaking, a 6dB improvement in the link budget will double the range in an LOS environment (8dBi gain antenna compared to a 2.1dBi gain di-pole antenna). To double the range in non-line of sight conditions it takes at least 12dB of additional receive sensitivity or transmit power but 12dB of additional antenna gain may not double the range because of the narrower antenna beamwidth which causes the antenna to ignore signals arriving from angles outside of the beamwidth.