Ah, okay. In that case, here are a few attempts to ground the idea philosophically:
It's the "prior-free" estimate with the least error. See that unbiased "prior-free" estimates must be mixtures of the (unbiased) estimates, and that biased estimates are dominated by being scaled to fit. So the best you can do is to pick the mixture that minimizes variance, which this is.
It actually is the point that maximizes the product of likelihoods (equivalently, the joint likelihood, since the estimate errors are assumed to be independent). You can see this by remembering that the Normal pdf is the inverse exponential quadratic, so you maximize the product of likelihoods by maximizing the sum of log-likelihoods, which happens where the log-likelihood slopes are each the negative of the other, which happens when distances are inversely proportional to the x^2 coefficients (or the weights are inversely proportional to the variances).
There's a pseudo-frequentist(?) version of this, where you treat each estimate as an assembly of (higher-variance) estimates at the same point, notice that the count is inversely proportional to the variance, and take the total population mean as your estimator. (You might like the mean for its L2-minimizing properties.)
A Bayesian interpretation is that, given the improper prior uniformly distributed over numbers and treating the two as independent pieces of evidence, the given formula gives the mode of the posterior (and, since the posterior is Normal, gives its mean and median as well).
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