In what episode does someone explain what a Section 8 actually means to Klinger?
There is no episode in which the consequences of a Section 8 are elucidated to Klinger. I think the idea is supposed to be that to a guy like Klinger, who clearly doesn't run with a crowd that places a ton of value on social responsibility and civic duty, a Section 8 based on a clever false ruse (wearing women's clothes, pretending to have sick family, etc.) is not actually grounds for negative judgment. In fact, he clearly believes it to be a common-sense street kid's reaction to dangerous duty he never wanted or volunteered for, and it's likely his peers and family would think about it the same way. Given his plans to run a restaurant, I don't think it'd affect his future plans, either.
On the other hand, what Sidney offers him is actually a very different thing, a general (or less-than-honorable) discharge due to basic incompatibility with service on account of homosexuality. That's precisely the kind of rap Klinger could never deal with, it'd shatter his respect in the Toledo pool hall crowd and brand him a lowly queer for the rest of his days. Klinger doesn't want to get out at any cost, just on terms that he can accept without shame. Crazy due to crossdressing can be disproven easily in the civilian world ("Am I wearing a dress now, bub?"), but the gay label is MUCH harder to shake off. Additionally, it's worth noting that a Section 8 is considered a Medical Discharge, aka equal to an Honorable Discharge. That's true today, as well. You still get all your normal benefits and can say you were discharged under honorable circumstances. A gay discharge is conspicuously different in the 1950s; you can't get a Dishonorable for anything short of getting convicted at court martial, but it still counts as "Less-Than-Honorable" and thus is much less of a desirable outcome.
Sidney's point was that if Klinger really wanted to get out badly enough, he had an open door, but he'd pay a lifelong price for it. Klinger wasn't interested in Toledo at any price, he wanted an easy out.
