From Bott and Tu's Differential Forms in Algebraic Topology:
Two submanifolds $R$ and $S$ in $M$ are said to intersect transversally iff $$T_xR + T_x S = T_x M$$ for all $x \in R \cap S$. For such a transversal intersection the codimension in $M$ is additive: $$\text{codim } R \cap S = \text{codim } R + \text{codim } S$$ This implies that the normal bundle of $R \cap S$ in $M$ is $$N_{R \cap S} = N_R \oplus N_S$$
I don't understand how this implication works. I understand it intuitively, but cannot see how it follows from the codimension thing. Any help would be appreciated.
I'll assume that normal bundles are defined as quotient objects, so for example $(N_R)_x = (T_x M) / (T_x R)$. From this it is clear that the injections $T_x(R \cap S) \hookrightarrow T_x R \hookrightarrow T_x M$ induce an injection $(N_R)_x \hookrightarrow (N_{R \cap S})_x$. Similarly, $(N_S)_x \hookrightarrow (N_{R \cap S})_x$. So, we may abuse notation and identify $(N_R)_x$ and $(N_S)_x$ with their embedded images in $(N_{R \cap S})_x$.
Furthermore, using the equation $T_x M = T_x R + T_x S$, together with a little algebra, we can deduce that $$(*) \qquad (N_{R \cap S})_x = (N_R)_x + (N_S)_x $$ (and it is here that I am using the notational abuse).
Next, we have $\text{codim} R \cap S = \text{dim} N_{R \cap S}$; and similarly for $R$ and for $S$. It follows that $$(**) \quad \dim N_{R \cap S} = \dim N_R + \dim N_S $$ Combining $(*)$ and $(**)$ we get $$(N_{R \cap S})_x = (N_R)_x \oplus (N_S)_x $$
Added: Here are some lemmas from linear algebra (the theory of vector spaces) which I used in deducing (*). You should be able to find these in any advanced linear algebra textbook.