Well essentially the price of any crypto goes up because more people want to buy some than sell some. Let's say 5% of people that get any hodl it, as time progresses there is a smaller and smaller supply. That's reason 1 for any crypto going up, including BAT.

The bigger the market place / the more people trade in / are involved in any crypto, the more people just kinda sit on a small amount without even really thinking about it. $5-10 here or there... BAT should have millions of users so that adds up.

Those are just general points that apply to any crypto.

BAT specifically; for anyone to earn BAT from watching ads, some advertiser has to actually buy BAT. That's FIAT direct to BAT transfer. That is money directly into the BAT economy from an external source. Those BAT have to come from somewhere and the more scarce the BAT is, the higher the price will be.

But yeah, essentially it's just the same as any crypto - the more it is used, the higher the price gets because demand out-strips supply.