Tensor Product in Quantum Computation
A tensor product of operations, $I\otimes J$ say, acts on each subsystem separately: if $\phi$ and $\psi$ are states and $I$ and $J$ are operators then
$$(I\otimes J)(\phi\otimes \psi) = (I\phi) \otimes (J\psi)$$
In bra-ket notation the state $\phi\otimes \psi$ can be denoted $|\phi\rangle|\psi\rangle$. 
In your first equation, the $\langle i|0\rangle\langle 0|k\rangle$ factor and $\langle j|I|l\rangle$ factor just separate in this way.

The algebra behind the second equation is basically:
$$(1+x^{2^0})\dots(1+x^{2^{t-1}})=1+x+x^2+x^2+\dots+x^{2^t-1}$$
except that the "1" is replaced by $| 0\rangle$, and the $x$ is replaced by $\exp(2\pi i \phi |1\rangle)$ (clash of my notation: $\phi$ is now a number). The only difference is that the multiplication is really a tensor product, and with bosons $|1\rangle\otimes |1\rangle=|2\rangle$.