In medical terms, something that is inflammatory causes a local reaction that shows up by swelling, heat, pain, and redness. If you are stung by a bee, you may have an acute inflammatory reaction to the sting.
The noun inflammatory also refers to something that excites anger, violence, rebellion, or similar strong emotions. In this case, the noun usually refers to speech or writing. If you are a politician, inflammatory speeches may be one tool you use to incite your constituents to vote against the corruption of your opponent. After all, a stirring speech about rights people may lose if an election goes a certain way is more likely to get out the vote than a more understated discussion.