Here is an attempted explanation. Unfortunately, this answer does not come with pictures, so you will have to draw them for me.
We have the point $P$, say $(1,1)$. We want to draw a line through $P$ with a given slope. I will give somewhat different instructions depending on whether the slope we are interested in is positive or negative or zero. Slope $0$ is very simple. Just draw the line through $P$ parallel to the $x$-axis.
Positive slope: Let the slope we want be say $\frac{2}{3}$. Go to the right of $P$ by $3$ units, then up by $2$ units. Let $Q$ be the point you reach. In our case, $Q=(4,3)$. The line through $P$ and $Q$ has slope $\frac{2}{3}$.
Let's check with algebra. The line through $(x_1,y_1)$ and $(x_2,y_2)$ has slope $\frac{y_2-y_1}{x_2-x_1}$ (the change in $y$ divided by the change in $x$.) So the line through $(1,1)$ and $(4,3)$ has slope $\frac{3-1}{4-1}=\frac{2}{3}$. That's what we wanted. For the rest of this post, we will concentrate on the geometry.
Let's practice a bit. Let $P$ be the point $(1,7)$. We want to draw the line through $P$ that has slope $\frac{5}{3}$. Go to the right by $3$. We reach $(4,7)$. Go up by $5$. We reach $Q=(4,12)$. The line $PQ$ has slope $\frac{5}{3}$.
Now again let $P=(1,7)$. We want to draw the line through $P$ that has slope $3$. Same idea, except that we must write the slope $3$ as $\frac{3}{1}$. Now right by $1$, up by $3$. We reach $Q=(2,10)$.
Negative slope: Suppose we want to draw the line through $P=(1,1)$ that has slope $-\frac{2}{3}$. From $P$, go left by $3$, up by $2$. We reach $Q=(-2,3)$. The line $PQ$ has slope $-\frac{2}{3}$.
Now for something fancy. Let $P=(1,5)$. Draw the line through $P$ that has slope $-\sqrt{3}$. Write the slope as $-\frac{\sqrt{3}}{1}$. Remember, left by $1$, up by $\sqrt{3}$. We reach $Q=(0,5+\sqrt{3}$. The line $PQ$ has slope $-\sqrt{3}$.
One needs practice, and a feeling for what lines of various slopes look like. Lines with positive slope go upwards. Lines with small positive slope, like $\frac{1}{10}$ go upwards kind of slowly, at a small angle. Think of a gently rising road. Lines with big positive slope, like $\frac{9}{2}$ go up fast. Think mountain climbing.
Lines with negative slope go downwards. If the slope is not very negative, like $-\frac{1}{7}$, the line goes down kind of slowly. If the slope is large negative, like $-\frac{31}{3}$, the line goes down steeply.