The metaphor cited below about the rising sea softening up the nut for you is a very apt way of characterizing Grothendieck's way of thinking.
In the particular example that your quote tries to explain, he tackled the following problem: In the geometry of the complex numbers, there is a good way of thinking about 'small' neighborhoods of a point, for instance all points that are fixed very small distance ('epsilon') away.
Unfortunately, this doesn't make sense algebraically: there is no polynomial equation that codifies being a 'small' distance away. What Grothendieck did was to essentially define away the problem by defining algebraic 'neighborhoods' (étale neighborhoods) that recover the complex analytic notion. These are no longer actual neighborhoods (they're not subsets) but rather spaces mapping to your original one.
The key insight here is that neighborhoods don't have to be a subspace of the space you start with. Once you allow your mind this freedom, then everything works out (with quite a bit more work of course!)
In the particular example that your quote tries to explain, he tackled the following problem: In the geometry of the complex numbers, there is a good way of thinking about 'small' neighborhoods of a point, for instance all points that are fixed very small distance ('epsilon') away.
Unfortunately, this doesn't make sense algebraically: there is no polynomial equation that codifies being a 'small' distance away. What Grothendieck did was to essentially define away the problem by defining algebraic 'neighborhoods' (étale neighborhoods) that recover the complex analytic notion. These are no longer actual neighborhoods (they're not subsets) but rather spaces mapping to your original one.
The key insight here is that neighborhoods don't have to be a subspace of the space you start with. Once you allow your mind this freedom, then everything works out (with quite a bit more work of course!)