What does 'infinitesimally small' mean?

"Infinitesimal" may be the opposite of "infinite", but it does not indicate any notion of logical negation. So infinitesimally small does not indicate a large object: it just emphasizes the smallness.

"Infinitesimally large" is not a very good phrase, and I would avoid it unless I wanted to play with irony (there is a conflict between the notions of 'infinitesimal' and 'large'). — Edited to add: as Roger points out in the comments, one can use "infinitesimally larger" to describe that one thing is larger than another, but only by an extremely small amount; here the fact that it's a comparison between two objects makes it a useable phrase (it's the difference in sizes which is small, not necessarily the 'largeness' of the objects themselves).

"Infinitely small", while perhaps not uncommon, is not quite as graceful as infinitesimally small. But not everyone knows the word 'infinitesimal'; as a computer scientist, I would only use it among physicists, computer scientists, mathematicians, and people whom I believe to have a large vocabulary. However, as with "infinitesimally larger", the phrase "infinitely smaller" is a very good phrase if you want to indicate that the sizes of two things are very different, and you wish to emphasize the smallness of one compared to the other.

Actually, infinitesimal means "very, very small".

Hence,

infinitesimally small

means very, very smallishly small, which, though it may be a pleonasm, very understandable.

infinitesimally large

means very , very smallishly large.

And that, in no way conceivable to at least my poor brain, makes any sense.

Or indeed, it makes about as much sense as "largely small" or "heavily light".

Since neither "large" nor "small" are absolute positives or negatives, they do not "cancel" each other "out", just like a white and black would not when applied to the same object.

Niel de Beaudrap's answer has discussed the fact that there is no logical negation going on. A separate issue is that "infinitesimally small" can be applied not just to a number but instead to a thing to be measured, and small versus large size is only one type of measurement. For example, a banker's heart could be infinitesimally small, a socialite's infinitesimally heavy. The former says you'd need a microscope to see it. The latter says that she spent an entire 7 seconds today pondering the plight of Romanian orphans. Things can be infinitesimally important, infinitesimally valuable, and so on.

In a purely mathematical context, there is a longstanding tradition of describing things by their measure. For example, Euclid says that a certain triangle and a certain square are equal, meaning that their areas are equal. If we feel that this blurs distinctions, then we can use a word such as "small" that clarifies that we consider the two shapes equally small in measure, not equal in the sense of being congruent or constituting the same point-sets.

Since one answer has claimed that "infinitesimally small" is a pleonasm, here's an example that wasn't written by ignorant people or for ignorant readers:

Only for an infinitesimally small region of four-dimensional space, i.e., for one in which the potentials g_μν can be considered a constant -- is 'velocity' defined at all.

--Janssen et al., The Genesis of General Relativity, p. 403. This is a good example of the use of "small" to clarify the meaning. Suppose the authors had written simply:

Only for an infinitesimal region is 'velocity' defined at all.

That would have left us wondering what measurement they had in mind for the region that would make its measure infinitesimal.

Googling for examples of "infinitesimally small," I also noticed that the phrase can come up in translation.

Cette difficulté peut être levée en supposant la différence de température entre le corps A et le corps B infiniment petite[...]

--Carnot, Reflexions sur la puissance motrice du feu, p. 27. Some translators render "infiniment petite" as "infinitesimally small," when "infinitely small" would have been more literal, and "infinitesimal" more idiomatic.

An argument against "infinitely small" is that "infinitely" doesn't literally mean "very, very," it means "enendingly." If a road is infinitely long, it means that you can walk on it for as long as you like, but you won't reach the end. That doesn't quite logically make sense when you apply it to something very, very small -- a small thing isn't "unendingly small" (which is what infiniment petite means).

In the sense that flammable and inflammable both mean liable to catch fire, we often see a redundant phrase added so infinitesimal means too small to be significant, as does infinitesimally small. Infinitessimally large would be an oxymoron (i.e. a contradiction in terms). Another example would be 'secret ballot' = 'secret secret vote' or 'black panther' = 'black black leopard'. Most examples are generated by the news media, not out of ignorance but in the belief their readers will not get the point otherwise