Understanding implemetation of the Toffoli gate using other gates
When applying quantum operations you always:


Simplify everything down to individual self-contained cases.
Apply the operation separately to each case.
Recombine the answers.


So, in the case of applying a CNOT to the state $|1\rangle \otimes (H \cdot |1\rangle)$, you:


Simplify things. We don't want that $H$ in the way. Apply it. The state is $|1\rangle \otimes (\frac{1}{\sqrt 2}|0\rangle - \frac{1}{\sqrt 2}|1\rangle)$.
Separate things. Distribute the tensor product over the subtraction. The state is $\frac{1}{\sqrt 2}|1\rangle \otimes |0\rangle  - \frac{1}{\sqrt 2}|1\rangle \otimes |1\rangle$. Or, for short, just $\frac{1}{\sqrt 2}|10\rangle  - \frac{1}{\sqrt 2}|11\rangle$.
Apply the operation separately to each case. If the second bit is our control, then in $|10\rangle$ the control isn't satisfied and nothing happens while in $|11\rangle$ the control is satisfied and we toggled to $|01\rangle$. Apply that to each part of $\frac{1}{\sqrt 2}|10\rangle  - \frac{1}{\sqrt 2}|11\rangle$ and you end up with $\frac{1}{\sqrt 2}|10\rangle  - \frac{1}{\sqrt 2}|01\rangle$.
None of our output cases overlapped when being added together, so there's no need to combine anything.


You might get a sense of this by adding state displays to Quirk while it cycles through the inputs into your Toffoli circuit.