How does it benefit the Stratis network if STRAT increases in value?

Stratis is the fuel for the Stratis Platform, a suite of products developed by Stratis Group Ltd. It is important for the security of the cryptocurrency that the price of STRAT goes up since Stratis is Proof of Stake. In order to mount a 51% attack on Stratis, the attacker would need to be staking 51% of the network weight. Right this minute, that would mean they would need to stake with over 20m STRAT. The higher the price of STRAT, the harder it gets to mount an attack.

Stratis Group Ltd were allocated 8m STRAT for development. You can see the breakdown of the post-ICO STRAT allocation here. The higher the price of STRAT, the more money Stratis Group Ltd have to fund development. Stratis Group Ltd have built the services of the Stratis Platform in such a way as to introduce value to STRAT when they are used.