The context is algebra, and I think the message is that you can improve your insight by thinking in terms of reciprocal values of various quantities that you normally use.

The context is actually mathematics in general, not specifically algebra. It's important to remember first of all that Jacobi was a 19th-century mathematician, and in those days mathematics didn't exactly have the same subdivisions it has now, and even to the extent it did, it was common for people to work across these boundaries. The quote at the beginning of this post refers to Jacobi as an algebraist for some reason, but you could equally well call him an analyst, for example.

The actual context of Jacobi's maxim is his work on elliptic functions, which he invented by inverting so-called "elliptic integrals". The inverses turned out to be much easier to work with. For an analogy, imagine that you'd never heard of the trigonometric functions sin and cos, but instead were working with arcsin and arccos, thinking of them as antiderivatives of and respectively. Along comes Jacobi (or someone in his role) who suggests considering the inverses of these things, giving them the names sin and cos, and shows how they satisfy all kinds of nice properties.

My own use of Jacobi's quote in Inverse Speed was meant to be slightly tongue-in-cheek, since the context I applied it to was so much more elementary than its original context. However, I felt that it was also a very illustrative application of the principle, its elementary nature notwithstanding. (Not to mention that a major subtext of my post was that I think of "elementary" concepts in terms of "advanced" concepts, since the latter are actually more natural to me.)

I see; I thought it was meant to be an example of a quote that sounds rational but isn't.