How does Cramer's rule work? I know Cramer's rule works for 3 linear equations. I know all steps to get solutions. But I don't know why (how) Cramer's rule gives us solutions? Why do we get $x=\frac{\Delta_1}\Delta$ and $y$ and $z$ in the same way? I want to know how these steps give us solutions?

It's actually simple; I explain it here in two variables, but the principle is the same.

Say you have an equation $$\begin{pmatrix}a&b\\\c&d\end{pmatrix}\begin{pmatrix}x\\\y\end{pmatrix}=\begin{pmatrix}p\\\q \end{pmatrix}$$

Now you can see that the following holds

$$\begin{pmatrix}a&b\\\c&d\end{pmatrix}\begin{pmatrix}x&0\\\y&1\end{pmatrix}=\begin{pmatrix}p&b\\\q &d\end{pmatrix}$$

Finally just take the determinant of this last equation; $\det$ is multiplicative so you get $$\Delta x=\Delta_1$$