Let's do it by example. Say we want the slope-intercept form of the line through $(1, 7)$ and parallel to $y = 3x - 6$. Slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept; so we just want to find the slope and the $y$-intercept of this new line.
The word "parallel" means "having the same slope", when we're talking about lines. So the slope of our line has to be the same as the slope of $y = 3x - 6$. Conveniently, $y = 3x - 6$ is already written in slope-intercept form - so we can just read off the slope, which is $3$. So our line should be $y = 3x + b$, where $b$ is the $y$-intercept.
Now, we could hunt down the $y$-intercept - or we could just make use of the fact that $(1,7)$ is on our line. To say "$(a,b)$ is on the line" is the same as saying "when you plug in $a$ for $x$, $y$ is $b$". So when we plug in $1$ for $x$, $y$ must be $7$. So, $7 = 3 \cdot 1 + b$. Solving for $b$ gives us $4$, so our answer is $4$.
I should say, there's another option - we could also make use of point-slope form. If you're not familiar with point-slope form, stop reading, because the next paragraph won't be helpful to you.
Point-slope form is $y - y_0 = m(x - x_0)$, where $(x_0,y_0)$ is a point on the line and $m$ is the slope. In this case, we knew that the slope was $3$ and $(1,7)$ was on the line, so the point-slope form of the line was $y - 7 = 3(x - 1)$. Distributing the $3$, we have $y - 7 = 3x - 3$. Adding $7$ to both sides, we have $y = 3x + 4$. This is slope-intercept form, so we're done (again).