To expand on Emerton's answer: Using the excision sequence, Cartan's result in the algebraic case boils down to showing the following: Let $R$ be a regular local ring, and $I$ and ideal of height at least $3$, then $H^i_I(R)=0$ for $i\leq 2$.This follows because: $$H^i_I(R) = lim \ Ext^i(R/I^n,R)$$
And $I^n$, being height $3$, always contain a regular sequence of length $2$, so the $Ext^i$ vanishes for $i\leq 2$ by standard result (see Bruns-Herzog Cohen Mcaulay book for example)
Incidentally, the hard and interesting question is to find upper bound for the vanishing of local cohomology, in other word, the cohomological dimension of a subvariety $Z$. Many strong results have been obtained after SGA, by Hartshorne, Ogus, Faltings, Huneke-Lyubeznik, etc. All the references can be found in Lyubeznik's paper (they were mentioned in the very first page).