A balanced mixer is one that suppresses the local oscillator (LO) input in the output. A double-balanced mixer (like the one you show here) suppresses both the RF input and the LO in the output. This is often accomplished with balanced transformers, hence the name.
Because T1's secondary has a grounded center tap, the voltages on its two "hot" output terminals are always equal and opposite in polarity with respect to ground. Assume the LO has driven T1's top output (+) and the bottom output (-). This forward biases diodes D1 and D2, and because T1's center tap is grounded, the diodes effectively clamp the lower side of T2's secondary to ground. Diodes D3 and D4 remain off.
Holding that state, now consider a second signal fed to T2's primary. Because the lower line of T2's secondary is clamped to ground, the RF input will appear at the IF output "right side up", i.e., in phase; if the RF input is (+), then so is the IF output.
Now reverse the polarity of the LO input to make the bottom output of T1 (+) and the top (-). Diodes D3 and D4 conduct while D1 and D2 remain off, clamping the top of T2's secondary to ground so that the RF input to T2 appears inverted at the IF output, i.e. 180 degrees out of phase.
But if there's no signal into T2, then the LO input to T1 does not appear at the IF, and by symmetry if there's no LO input signal into T1 then the RF signal into T2 also does not appear at the IF output.
Mathematically, mixing is multiplication. When you multiply two numbers with the same sign (both positive or both negative), the result is a positive number. When the operands have opposite signals (one positive, one negative), the output is negative.
So this circuit multiplies the LO and RF signals and their product appears at the IF output. This is why these mixers are sometimes called "product detectors" when used to detect, say, SSB or CW.
This circuit is a way to multiply two analog signals with simple passive components. The same operation can be done with an exclusive-OR logic gate if a positive input is mapped to a logic '1' and a negative input is mapped to a logic '0'.