Worn tires are more suspectible to flats due to the thinning of the tread rubber.
In particular, driving on wet roads becomes very dangerous with loger stopping distance.
The international standard for safe tread depth is 1.6mm although countries do set their own legal limits.

Tread Wear Indicator

Tires have built-in tread wear indicators which mark the minimum allowable tread depth. The indicators are 4~6 small raised bars which run across the grooves of the tire tread and are 1.6mm in height. Made to be clearly visible to consumers, the mark in the shoulder area also helps to pinpoint the location of the indicators. When measuring tread depth, one should avoid the area with the indicators.

Tires with worn out treads lack the ability to channel water, creating extremely dangerous situations such as slipping or hydroplaning on wet roads. Tire blow outs also increase in frequency.