The following is a historical question, but first some background:
Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be shown to be equal (even for an infinite dimensional vector space):
$$ \det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right) $$
Here is a simple example using the associated matrix in the finite dimensional case:
For more on this topic (and how to ensure the definitions make sense for infinite dimensional vector spaces) see: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer-Verlag, New York, 1984.
In particular, see Koblitz's book for how ($p$-adic versions of) this identity can be used in Bernard Dwork's proof of the rationality of the zeta function (resolving the first of the Weil Conjectures).
I would welcome any insight as to how one thinks up, observes, or conjectures such an identity in the first place, but my question is: what is the origin of this identity and where else has it appeared (perhaps in a somewhat modified form)?
Using $\det\exp=\exp\mathrm{tr}$, we have $$ \det(I-tA) = \det \exp\log(I-tA) = \exp\mathrm{tr}\log(I-tA) $$ and as $$ \log(I-tA) = -\sum_{k\ge1} \frac1k t^k A^k, $$ we have $$ \exp\mathrm{tr}\log(I-tA) = \exp\left( -\sum_{k\ge1} \frac1k t^k \mathrm{tr}(A^k)\right) $$ This shows that your identity is a form of the $\det\exp$ identity.
The question about the origins is interesting and I do not have much to say (which was why I restricted myself to a comment in the first place). For complex matrices it is very easy to prove that $\det\exp(A)=\exp\mathrm{tr}(A)$, which might explain why I have never seen a name attached to it.
I have seen it used in connection with Lie groups (to prove that sl is the Lie algebra of SL), and (in forms like what you stated) in work on enumeration.