First of all, note that T1 and T2 are both PNP transistors. To turn a PNP transistor on, the base must be more negative than the emitter.
When the circuit first powers up, T1 is off, and T2 is turned on by the current through R3 and R4.
As more current is drawn by the load (R5), the voltage drop across R1 increases, making the base of T1 more negative with respect to its emitter. When this voltage drop reaches about 1.3V — the forward voltage drop of T1's B-E junction plus the forward drop of D — current begins to flow through D and T1, turning T1 on. This reduces the base drive to T2, causing it to start to turn off.
As T2 turns off, the voltage across the load drops, causing current to flow through R2. Since this current can't flow through D, it must flow through T1, turning it on harder, and turning T2 off more. This process continues until T1 is saturated, T2 is completely cut off because there's no voltage across R3, and there's only a trickle of current flowing through T1, R2 and the load. The collector current of T1 is flowing through R4 to ground. Now the fuse is "tripped". The load (or the supply) must be removed altogether in order to reset the fuse and restore power to the load.