Why do people use anti-inflammatory drugs?
You are correct in that inflammation is a physiological process. It is vital for the correct development and localisation of a strong immune response, and contributes to tissue healing.

However, the beneficial effects are replaced by dangerous processes when they are prolonged or too intense. This includes both effects of leukocyte attraction and invasion, but also deleterious consequences of the accumulated cytokines. For example, TNF-alpha originally contributes to immune cell activation but in prolonged inflammation prevents tissue healing.

Note that "prolonged inflammation" does not necessarily mean that inflammation has been apparent or noticeable for the affected person for a long time. Often by the time symptoms present, the inflammation process has been persistent and become pathological. At this point, anti-inflammatory drugs need to be used as soon as possible to minimise irreversible damage.

Aside from this, quite often inflammatory and immune processes are triggered without a need for them. Our immune system is quite prone to errors, and cases of misguided immune reactions are actually increasing. Consider for example allergies, which often manifest with skin-symptoms such as eczema that can be treated with anti-inflammatory creams.