When #T > T_f#, i.e. when the surroundings are hotter than the freezing (melting) point (if #T = T_f#, that is the condition for melting or freezing, but neither one is more likely than the other).
When that is the case, there is an ongoing (and spontaneous) attempt by the surroundings and by the ice to achieve thermal equilibrium, and that requires getting past the "barrier" of melting the ice (approximately #"6.02 kJ/mol"#, the enthalpy of fusion).
Once that "barrier" is overcome, the ice can melt and warm up to the hotter temperature #T#, so that the temperature of the newly-formed liquid water becomes equal to the temperature of the surroundings (achieving thermal equilibrium).
The larger #T - T_f# is, the faster the melting occurs, because the more thermal energy is present that can absorb itself into the ice.