Ironing of clothes is a physical change (if you believe that physical changes and chemical changes are not part of a single continuum of change).
Most polymers (including the cellulose fibers that comprise cotton) have a glass transition. The glass transition is a reversible phase change from a rigid fixed solid amorphous "glass" to a solid phase where there is some mobility of the polymer chains, but not enough to be a liquid (this phase is sometimes called a "rubber," "plastic," or "elastomer" phase). The glass transition is sometimes used to refer to the phase change between an order crystalline (or semicrystalline) solid to a disordered amorphous solid.
Above the glass transition temperature, the solid material has some mobility - it can be shaped, molded, flattened, whatever. When it is cooled down again, it becomes rigid and holds its shape.
Ironing raises the temperature of the fabric to above the glass transition. The action of ironing presses out the wrinkles in the now more flexible fabric. When the fabric cools, it retains its new wrinkle-free shape.
Water is used during ironing for one important purpose. I tried to find the glass transition temperature of cellulose, but am unable to do so. Presumably it is close to the temperature at which cellulose begins to burn. Water prevents the burning, but not just in the way you would expect. Water does absorb heat from the iron in order to keep the fabric from getting too hot. However, if this were the only thing water was doing, the fabric would not reach the glass transition and ironing would not work. Water also acts as a plasticizer, lowering the glass transition temperature. Most fabrics are hydrophilic (and cellulose as a carbohydrate is especially so). The water binds to the chains, swells the fibers, and changes the physical and chemical properties of the material.