The long play is that it actually doesn't cost significant money per trade at an institutional level. The only reason the other major brokers charge you money is because it covers their brick and mortar retail costs (e.g. Scottrade) and it adds to the bottomline (e.g. Etrade). So Robin Hood is actually just giving this surplus to the consumer instead of keeping it for themselves, and building a user base / brand in the process. Eventually once they have enough users, they will start to up sell you for other services (e.g. checking/savings, margin trading, options trading, check writing etc).
So majority of the brokers have brick and mortar operations that can't afford to compete on price. That leaves only the online brokers. Then the next step is to build a brand that attracts customer trust/loyalty, which is what Robinhood seems to be doing and to build a really nice experience which they do with their slick apps. Once customers (generally young ppl who are first time investors) are invested in the Robinhood platform, then Robinhood can up sell them on additional features/services (margin, options, API access etc). So sure Etrade could compete on price (I doubt they would want to since it would immediately hit the bottom line vs a new company not depending on broker fees), but if you look at it from a long term perspective eliminating broker fees is just step 1.
