Let $f$ be a Schwartz function on the ring of adeles $\mathbb{A}$ of a number field $K$, and $d^\times x$ the multiplicative Haar measure on $\mathbb{A}^\times$. One can embed $K^\times$ diagonally in $\mathbb{A}^\times$ and can take a quotient $\mathbb{A}^\times/K^\times$. In Tate's thesis, the integral of $f$ over $\mathbb{A}^\times$ is replaced with an integral over $\mathbb{A}^\times/K^\times$ via
$$\int_{\mathbb{A}^\times}f(x)d^\times x = \int_{\mathbb{A}^\times/K^\times}\left(\sum_{\kappa \in K^\times}f(\kappa x)\right) d^\times x.$$
I am wondering how to interpret this sum. In particular, why does it converge, and how big is $\mathbb{A}^\times/K^\times$? For example, if in the real archimedean component $f_\infty(x) = e^{-\pi x^2}$, is he integrating over a function which in this component is equal to $$\sum_{\kappa \in K^\times}e^{-\pi \kappa^2 x^2} $$ and does this converge?
Also, how does this relate to the Jacobi theta function?
You need to work out an actual example so you can see that some of what you wrote makes no sense.
Take $K = \mathbf Q$. In the definition of Schwartz functions on $\mathbf A_\mathbf Q$, they are sums of finitely many products of (nice) local functions with most $p$-factors being the characteristic function of $\mathbf Z_p$. The simplest example is the product function $f(\mathbf x) = e^{-\pi x^2}\prod_p \chi_{\mathbf Z_p}(x_p)$, where $x = x_\infty$.
Fix an idele $\mathbf x$ in $\mathbf A_\mathbf Q^\times$. Let's see why the series $\sum_{k \in \mathbf Q^\times} f(k\mathbf x)$ converges and what it equals. We'll see it is not $\sum_{k \in \mathbf Q^\times} e^{-\pi k^2x^2}$, a sum over all nonzero rationals that converges for no real $x$.
Set $S = \{p : x_p \not\in \mathbf Z_p^\times\}$, which is finite. Let $r = \prod_{p\in S}p^{v_p(x_p)}$, so $r$ is a positive rational number depending on the idele $\mathbf x$ and $x_p/r \in \mathbf Z_p$ for all $p$ (not just $p \in S$).
When $k\in\mathbf Q^\times$, $f(k\mathbf x)=0$ unless $kx_p \in \mathbf Z_p$ for every prime $p$, due to how $f$ is defined. Rewrite "$kx_p \in \mathbf Z_p$" as $(kr)(x_p/r) \in \mathbf Z_p$. Since $x_p/r \in \mathbf Z_p^\times$ for each $p$ (not just $p \in S$, but all $p$), $kx_p \in \mathbf Z_p$ if and only if $kr \in \mathbf Z_p$. We need this for all $p$, which is the same as saying $kr \in \mathbf Z$. In other words, the only nonzero rational $k$ for which $f(k\mathbf x)\not=0$ are the $k \in (1/r)\mathbf Z - \{0\}$. Write $k = n/r$ for a nonzero integer $n$, so (using the actual definition of $f$ as a product of local functions) $$ f(k\mathbf x) = f(n(1/r)\mathbf x) = e^{-\pi (nx/r)^2} = e^{-\pi n^2x^2/r^2} $$ since $(1/r)\mathbf x \in \mathbf R^\times \times \prod_p \mathbf Z_p^\times$. Note $r$ depends on $\mathbf x$. (This $r$ is the first factor in a nice direct product decomposition: $\mathbf A_\mathbf Q^\times = \mathbf Q_{>0} \times \mathbf R^\times \times \prod_p \mathbf Z_p^\times$. When $K \not= \mathbf Q$, $\mathbf A_K^\times$ does not have anything like that tidy decomposition, which makes general idele groups and Hecke characters complicated!)
Now for each idele $\mathbf x$ of $\mathbf Q$, with its associated positive rational $r$, we can write your mystery sum at $\mathbf x$ in the integral as something less mysterious: $$ \sum_{k \in \mathbf Q^\times} f(k\mathbf x) = \sum_{n \in \mathbf Z - \{0\}} e^{-\pi n^2x^2/r^2} = \sum_{n \in \mathbf Z} e^{-\pi n^2(x/r)^2} - 1, $$ which (ignoring the term $-1$ at the end) is the classical Jacobi theta-function at $(x/r)^2$, where $x$ is the archimedean component of the idele $\mathbf x$. In particular, this sum depends only on the archimedean component of $\mathbf x$ and it obviously converges if you know about Jacobi theta-functions.
I haven't actually dealt directly here with the integral, just showing that the sum inside the integral makes sense when we take $f$ to be the simplest nonzero Schwartz function. What one should really be integrating is not that function $f$ I wrote down, but the product $f(\mathbf x)||\mathbf x||^s$, where $||\cdot||$ is the idelic norm and ${\rm Re}(s)$ is large enough. It turns out that the integral with that function converges when ${\rm Re}(s) > 1$ and (for suitable idelic Haar measure) is exactly the Riemann zeta-function. This is worked out in Section 6 of Gelbart and Miller's paper "Riemann's Zeta Function and Beyond" here.