Minami in "K-groups of symmetric spaces" (equations 1.1, 1.2) states the following, originally due to Hodgkins:
Suppose that $G$ is a compact connected Lie group such that $\pi_1(G)$ is torison-free and let $K_G^*$ denote the equivariant K-theory associated with $G$, in [8],[9] Hodgkin constructed a strongly convergent spectral sequence (assuming one of X or Y is a free $G$-space) $$ E_2^{* \,*}= Tor_{R(G)}^{*\,*}(K_G^*(X),K_G^*(Y)) \implies K_G^*(X \times Y)$$
They go on to say that in the case $X = G$, $Y = G/H$ the spectral sequence becomes:
$$ E_2^{* \,0}= Tor_{R(G)}^{*\,0}(\mathbb{Z},R(H)) \implies K^*(G/H)$$
They then state a theorem (whose proof takes up a chunk of the paper) saying that the latter s.s. collapses when $H$ is a closed subgroup. I do not understand what he means by collapses, despite briefly reading through the proof and going to the referenced theorem of Snaith in his "Masssey products in K-theory". My thoughts:
Firstly, I believe the s.s. is $(\mathbb{Z}, \mathbb{Z}_2)$-graded, so differentials on odd pages map horizontally i.e $d^{2k+1}: E^{p,\,q} \rightarrow E^{p+2k+1,\,q}$, otherwise there is nothing to say about the latter s.s. This could be incorrect though.
$1$. The sequence collapses in the usual/modern definition of the word, i.e. the sequence lives in a single row or column. This is trivially true, unless they meant collapses into a single column (as well as a single row) but then this isn't true.
$2.$ All differentials from the $2^{nd}$ page onwards are zero, if this was the case I would expect the word degenerates to be used.
$3$. The sequence degenerates after a finite number of pages. Again I think this is trivial since any fixed $G$ has finite rank, since it's compact - so the pages can only have non-zero entries for so long, so eventually the differentials start mapping "into the void".
The only one which seems not trivial to me is option $2$, is this what was meant?
Yes, option 2 is what is meant. Equivalently, all classes on the $E_2$ page are permanent cycles: if you look at the proof of Lemma 2.2, you'll find this is what is really being proved. It is also compatible with the reference to Snaith, whose collapsing results also amount to showing that everything on the $E_2$ page is a permanent cycle.