The T1WI relies upon the longitudinal relaxation of the net magnetisation vector (NMV). T1 weighting tend to have short TE and TR times.
Fat has a large longitudinal and transverse magnetization, and appears bright on a T1 weighted image. Conversely, water has less longitudinal magnetization prior to a RF pulse, and therefore has less transverse magnetization after a RF pulse. Thus, water has low signal and appears dark.