When $X,Y$ are $k$- and $l$-manifolds, we can have a function $f:X\rightarrow Y, x\in X$ such that $f$ is an immersion resp. submersion at $x$. The local immersion/submersion theorem now says: There now are coordinates around $x$ such that $$f(x_1,\ldots,x_k)=\begin{cases} (x_1,\ldots,x_k,0,\ldots,0) &\text{in case of an immersion}\\ (x_1,\ldots,x_l) &\text{in case of a submersion}\\ \end{cases}$$
Now I don't really know what I can do with this theorems. In what way should I think about the results, how can I use them and is there an intuitive meaning of the theorem?