The cause of wrinkles in any material, including shirts is a result of hear and water. Like shrinking, the bonds are the problem and with this problem, the bonds are broken and this moves them into different positions.
An iron removes wrinkles in the same way, by heating up the fibres, new bonds will form that are no longer wrinkled.
Water is also a cause of wrinkles. The fibres have the same molecules as water that hold them together. Like heat, the water penetrates the fibres and create different bonds and form wrinkles.
To try to prevent wrinkles from happening don’t wear clothes that are freshly ironed or out of the dryer or damp as fabric crease easiest when they are warm or damp. Ensure they are dry and cooled down before putting them on.
There are anti-wrinkle sprays available and while they will not completely put a stop to wrinkles but will certainly help in the quest for wrinkle free clothing.
Wrinkles will look pretty terrible when you both go out in your coordinated shirts so have them washed and ironed in the above way and you can both spend the day looking awesome.