We do know that metastasis (spreading of the cancer) is clearly linked to inflammation, and aspirin reduces inflammation.
Of course, aspirin is not the only choice for reducing inflammation; other dietary and lifestyle habits greatly affect the level of inflammation throughout the body. We know about this particularly as affects the cardiovascular system, and that's the reason the study subjects were taking the aspirin in the first place.
These findings, from the well-known Nurses' Health Study, offer intriguing new evidence about the link between cancer and inflammation. Here is an excerpt from a lengthier mainstream press reporting on the study's findings:
http://www.latimes.com/news/nation-and-world/la-sci-aspirin17-2010feb17,0,4190955.story
"They found that women who took aspirin two to five days per week were 60% less likely to have a recurrence and 71% less likely to die from breast cancer. Those taking it more frequently had a 43% lower risk of recurrence and a 64% lower risk of death."
In research, it's always important not to confuse association with causation. In other words, perhaps there were other lifestyle factors among the nurses who took the aspirin that affected their rates of inflammation, cancer risk, etc. (It makes sense that the study subjects who were educated enough to consume a daily aspirin might know to take other steps to reduce inflammation.)
Another excerpt from the LA Times article:
"The researchers did not monitor aspirin dosage, but the majority of the women taking aspirin did so for its heart benefit, and thus were taking 81 milligrams per day, less than a normal dose. These women may have been more health conscious than other women in the study, and it is possible that other actions on their part were responsible for their improved survival, experts noted."
It's important to note that the Nurses Study is based on self reports.
Finally, there is this important caution in the article:
"Whether aspirin ultimately proves beneficial or not, doctors cautioned that women being treated for breast cancer should not take the medication. It can interfere with the therapy, producing severe side effects."
I hope that helps answer your question.
This simple drug, derived initially from willow bark tea centuries ago, has wide ranging effects that are still being elucidated.
The science of the process most likely has to do with what are called cyclooxygenase inhibitors (COX inhibition).