I found the following discussion about the geometric interpretation of regular sequences very helpful: What is a geometric interpretation of regular sequences in various instances?
However I tried to prove one of the statements, but couldn't quite solve it:
Let $R:=K[x_1,...,x_n]/I$ and $X:=V(I)$. $f_1,...,f_r$ is a regular sequence of the ring $R$, iff $f_i$ is non-zero on all the irreducible components of $V_X(f_1,...,f_{i-1})$ for $i=2,...,n$ and $f_1$ is non-zero in all the components of $X$.
I had some problems with the direction "$\Leftarrow$". Suppose $X_1,...,X_k$ are the irreducible components of $V(f_1,...,f_{i-1})$, $f_i(X_j)\neq 0$ for $j=1,...,k$ and $rf_i\in (f_1,...,f_{i-1})$ for some $r\in R$. That means that $$ rf_i\in \bigcap_{j=1}^k I(X_k)=\sqrt{(f_1,...,f_{i-1})} $$ and therefore $r\in \sqrt{(f_1,...,f_{i-1})}$, since $\bigcap_{j=1}^kI(X_k)$ is the decomposition of $\sqrt{(f_1,...,f_{i-1})}$ into prime ideals. So there is a $d\in \mathbb{N}$ with $r^d\in (f_1,...,f_{i-1})$. That's the point where I am stuck. We obviously want $d=1$. But I'm not sure how to progress. Maybe one should use the fact that $K[x_1,...,x_n]$ is an UFD and argue all the way from there?