Rashi explains this miraculously, with the clouds of Glory dry-cleaning the clothing and the clothes growing with them. I would point out now that this midrash plays on the word mei'alecha, that they did not even have to remove them to make them fresh, and the children never had to take them off because they outgrew them.
|ד שִׂמְלָתְךָ לֹא בָלְתָה, מֵעָלֶיךָ, וְרַגְלְךָ, לֹא בָצֵקָה--זֶה, אַרְבָּעִים שָׁנָה.||4 Thy raiment waxed not old upon thee, neither did thy foot swell, these forty years.|
Shadal says that they had plenty of changes of clothing, such that they never were wearing old, worn-out clothing. And Ibn Ezra considers this same idea, but based on Greek and medieval science and the pasuk labeling the manna as sublime food, determines that they did not sweat out the impurities, and thus spoil their clothing.
This year, DovBear linked to my post (thanks!), noting that this alternate explanation of Shadal ruins the "Torah-true" explanation for the absence of archaeological evidence, on the basis of the midrash. And there is discussion in the comment section there.
What about the second half of the pasuk? Why should having infinite changes of clothing stop the feet from swelling? Well, I omitted the second half of Shadal's commentary in the previous post. He writes:
Thus, he does not interpret batzeika as "swell". Rather, as "develop calluses". Since they had plenty of changes of shoes, they never had to go barefoot, and thus never developed hard calluses on the bottom of their feet. This then works out quite well with Shadal's explanation of the first half of the pasuk.
In terms of absence of evidence, as I note there, I am no expert. But one can envision all sorts of conditions which would cause the material from that time and place not to be preserved. We don't know how many such instances things have not been preserved but we do not know of it. Indeed, this seems tautological.
And for decades, archeologists asserted, based on absence of evidence, that there was no country of Edom before the 8th century, such that Biblical accounts of interactions with Edom in the time of David and Shlomo were ahistorical. And then they decided to dig elsewhere, and found the evidence. But if they had not looked in that particular place, or if the evidence in that other place was not sufficient or long-lasting, as in the other places, then they would have continued to scoff at the ahistoricity of the Biblical account of Edom at that time. My point is that we should realize that this is a reconstruction, and should not necessarily treat present or even future archaeological conceptions of Biblical history as absolute.