The statement about converting in a single ended voltage signal is simply wrong, which is probably causing all the confusion.
The constant current source sets up a total current I.
The sum of currents in T1 and T2 must be equal to I, so I1 + I2 = I;
We can also say I2 = I - I1;
The currents in T3 and T4 are equal to I1: I1 = I3 = I4;
The main thing is, the output is very high impedance (as it's basically just two collectors, hence current sources). So it delivers current, not voltage. More specifically, the output current will be the difference between I2 and I4.
On a perfectly balanced differential pair, the output current is zero, because I1 = I3 = I4 = I/2.
And I2 = I - I1 = I - I/2 = I/2;
Since I2 and I4 are equal, there is no "extra current" left for the output, hence the output current is zero.
With maximal imbalance, we get e.g. I1 = I3 = I4 = I;
And I2 = I - I1 = I - I = 0;
Now I2 is zero, but I4 = I, so the output sources a current equal to I.
The opposite would happen if I2 = I and I1 = 0, then the output sinks a current equal to I.
Because of the high impedance output, we can connect a high impedance load, where minimal current changes would give very high voltage changes, thus getting very high voltage gain. Even if the load impedance were the same as the resistor on the collector of T2 (in a diff. pair without current mirror), the gain would still be twice as much, because the signal of T1 is not wasted anymore, as the current mirror enables the output to not only sink current, but also source it.