In short, the series $K_phi$ is the "Hirzebruch characteristic series" which arises in the construction/calculation of genera, and in Hirzebruch-Riemann-Roch. The first few chapters of Manifolds and modular forms by Hirzebruch et al. describe the classical version of this pretty well.
If I have a one dimensional formal group law $F$ over a ring $A$, then over $A_{mathbb{Q}}$ there is an isomorphism $mathrm{exp}_F: G_ato F$ with the additive formal group. Let $K(x)=x/mathrm{exp}_F(x)$.
Now suppose $R$ is a "complex orientable cohomology theory", which means we are given a suitable isomorphism of rings $R^*(CP^infty)approx pi_*R[[x]]$. Such a theory has an associated formal group law $F$ (induced by the map $CP^inftytimes CP^inftyto CP^infty$ which classifies tensor product of line bundles), and thus there is an assocated series $K(x)=x/mathrm{exp}_F(x)$ in $pi_*R_mathbb{Q}[[x]]$.
It turns out that a map of ring spectra $phi:MUto R$ corresponds exactly to giving a complex orientation of $R$. By Thom, elements of $pi_*MU$ correspond to cobordism classes of stably-almost-complex manifolds, and there is a standard calculus due to Hirzebruch of calculating the effect of the map $pi_*MU to pi_*R_{mathbb{Q}}$ using $K(x)$, which I might as well call $K_phi(x)$, since it depends on $phi$. The formula (if I remember correctly), is that, if $[M]in pi_*MU$ is the class corresponding to a manifold of dimension $2n$, then
$$
phi(M) = langle K_phi(x_1)dots K_phi(x_n), [M] rangle,
$$
where the $x_i$ are the "chern roots" of the tangent bundle of $M$, and $[M]in H_{2n}M$ is the fundamental class.
There is a "universal example" of a $K_phi$, corresponding to the identity map $phicolon MUto MU$. It turns out that $pi_*MU_{mathbb{Q}}$ is a polynomial ring on the coefficients of $K_phi$, so that $K_phi(x)=sum a_{i-1}x^i$ (with $a_0=1$) and $pi_*MU_{mathbb{Q}}=mathbb{Q}[a_1,a_2,dots]$. (I'll need this later.)
In his talk, Mike isn't talking about complex orientations, but rather orientations with respect to $MSpin$ or $MOlangle 8rangle$ (instead of $MOlangle 8rangle$, we call it $MString$ these days, for some reason).
There is a map of ring spectrum $MUto MSO$, induced by the apparent homomorphisms $U(n)to SO(2n)$ of Lie groups. There is also a map $MSpinto MSO$, induced by the double cover of lie groups. Although $MSpinneq MSO$, we have that $pi_*MSpin_{mathbb{Q}}to pi_*MSO_{mathbb{Q}}$ is an isomorphsism. Thus, a map $phicolon MSpinto R$ induces
$$ pi_*MU_{mathbb{Q}}to pi_*MSO_{mathbb{Q}}approx pi_*MSpin_{mathbb{Q}}to pi_*R_{mathbb{Q}},$$
and we can get $K_phi(x)$ from this.
The $MOlangle 8rangle$ case is a little trickier.
There is a map $MUlangle 6rangle to MOlangle 8rangle$, so a ring spectrum map $phicolon MOlangle 8rangleto R$ gives rise to a map
$$ MUlangle 6rangle_{mathbb{Q}} to R_{mathbb{Q}}.$$
On the other hand, the effect of the map $MUlangle 6rangleto MU$, on homotopy groups tensored with $mathbb{Q}$, is
$$ mathbb{Q}[a_3,a_4,dots]to mathbb{Q}[a_1,a_2,a_3,dots].$$
So a map $phicolon MOlangle 8rangleto R$ gives us elements $phi(a_i)in pi_{2i}R$ for $igeq3$, which we can use as the coefficients of a series $K_phi(x)in pi_*R_{mathbb{Q}}$.
I must point out: there is actually an error in the statement of (2) and (3) given in Mike's talk. What he writes down are the "Kummer congruences"; but what one really needs to require are the "generalized Kummer congruences", which are basically the collection of all possible $p$-adic congruences involving Bernoulli numbers, not just the ones listed in (2) and (3). This comes from the theory of the "Mazur measure": the generalized Kummer congruences imply that the sequence $b_n(1-p^{n-1})(1-c^n)$ can be interpolated to a function $f$, so that $f(n)$ for $n$ an integer is the moment of a measure on $mathbb{Z}_p^times$. With (2) and (3) replaced by "interpolates to the moments of a measure on $mathbb{Z}_p^times$", the result is correct.
Finally: There is a writeup of this at http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf, which may or may not be of any use to you!
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