The purpose of mathematical models in both economics and finance isn't to be exactly right, but to structure discussion and assist in decision making.
Some models such as the Black-Scholes-Merton model in option pricing is still widely used by practitioners, in spite of the fact that it is known to be flawed in all matters of ways. However, it has the advantage of simplicity and of presenting problems that are well defined, well known and with which you can try to cope informally. It has also gifted us with a way to think about option prices: implied volatility is computed by asking which volatility level you would need for BSM to match observed prices exactly. This gave us the volatility smile and the knowledge that the slope and level of that beast changes with maturity and across time. This gives you a hint as to how you can improve on BSM: you need to find a way to put more probability density than BSM does in the zones where IV is high.
Now, moving back to the issue of causality, it is not entirely fair. Take again the example of option pricing. Ideally, what you would like to do is to reconcile the cross-section of option prices with the time series of returns of the underlying -- as Bates (1996) pointed out. Although many people force fit their option pricing model directly under the Q measure, caring only about getting the cross-section part right, more recent papers have tried to reconcile both sources of data. The link between both the risk-neutral and the physical measure is the pricing kernel or the stochastic discount factor: this beast soaks up concerns about things like risk aversion as it would invariably emerge in a general equilibrium model. That actually is an economic explanation, something structural which hinges on investor preferences.
This is even more explicit if you look at the more interesting option pricing models: all of them showcase incomplete markets, which means there is an infinite number of ways to eliminate risk from the model. In the earlier literature, people would pick a price for risk by motivating their choice from a general equilibrium model. That is what Heston (1993) did, even if it is a footnote seemingly no one remembers and this is what Duan (1995) did in the option pricing context. When you use the Heston model, for example, you're actually relying on an economic theory behind -- i.e., you know for a fact that there exist some equilibrium that supports your choice of measure. A more recent exception would be Christoffersen, Elkami, Fenou and Heston (2010) that proposed to slightly generalize the work of Duan (1995), but without relying on an equilibrium model. They do not have a formal theory behind the quadratic kernel used in Christoffersen, Heston and Jacobs (2013), but there is an empirical motivation (matching ratios of density estimates) and an informal economic intuition (people value volatility directly).
Clearly, there is a lot more going on here that simply describing statistical patterns. These models can be used to "predict the future," although it depends on what you mean by prediction. They are designed to capture intricate behavior and changes in the higher moments of conditional distributions. If you want to think about the likelihood of something bad happening, they are well suited to do just that. They aren't peculiarly interesting model of the conditional mean -- that's what people usually try to model and predict, at least as far as the squared loss they use in estimation and evaluation suggests.
