What the other answers seem to miss is that there isn't just a vague or qualitative relation between delta and the probability of expiring in the money. Though the two are not equal in general, there is a precise limiting case in which they converge (and so in many practical cases they may be very close). This is the case where the Black-Scholes assumptions hold and the expected relative changes in the underlying price by expiration are small enough to be linearized (e.g., the option is close to expiration). Then the lognormal distribution converges to a plain normal one.
We can then state Acccumulation's observation more precisely, say for a call. If the current underlying price moves $1 higher while the other parameters (including volatility) are unchanged, then the (normal) probability distribution of price at expiration shifts, with its mean, additively upward by $1. The current option value is the expectation of its value at expiration. The in-the-money values increase by $1 while the out-of-the-money values remain unchanged (worthless). By linearity of expectation, the change in option value (delta) therefore equals the in-the-money probability.
The two will diverge for longer-dated options on volatile underlyings, or when Black-Scholes assumptions are strongly violated.