Mathematically it seems like the expected rate of return, whatever that might be, is the same for both. An aggressive strategy is higher risk and higher reward. A conservative strategy is lower risk and lower reward.
That is not true. Roughly, the mathematical analogue of "higher risk and higher reward" is "higher standard deviation and higher mean". In other words, the aggressive strategy does have a higher expected rate of return (higher mean). Its disadvantage is that it has a higher likelihood of incurring intermediate losses (and/or higher magnitude of intermediate losses) on the way.
This is classically illustrated with the following chart -
from Vanguard. You can see that the average return is greater the riskier the portfolio (i.e., the more allocated to stocks relative to bonds), but this higher average return comes at the price of a greater range of possible returns. With an aggressive portfolio, you take a greater risk of losses at any given moment for a greater chance of gains over a long period.
Given this, it should be obvious why the advice is to be aggressive early on. Early in life, you don't care about whether your current position is up or down, because you're not taking the money out. If your portfolio is down, you just leave the money in there until it goes back up again. Later in life, you need to spend the money; you now care about whether your current position is up or down, because you can't afford to wait out a down market and may have to realize a loss by selling.
It's important to note that the expected return is always greater for a higher-risk portfolio, as is the expected risk; the expected rate of return doesn't magically change as you age. What changes is your ability to absorb losses to hold out for later gains.