Look at your pricing tree - even if the first period is down, there's a possibility that the next period is back up to $310, making the option worth $10 at that point. So there's a (roughly) 50% change of it being worth $10 and a 50% change of it being worth $0 after 2 periods, making that period's value about $5.
Only the last period in the binomial model uses intrinsic value - each of the nodes in the prior periods use a combination of the two possible values to the right of it (plus some discounting).
Binomial model aside, an option always has value, since the buyer pays nothing if the option expires out of the money, so there's no possibility of loss from the option itself. There's always a possibility (which may be very slim) that the stock goes back up and the option becomes ITM.
The higher the volatility of the stock, the higher that probability and the higher the option price, even if it's currently out of the money.