AND-gate transistor structure
First consider that floating inputs are not considered good design practice.  And such a configuration leads to unexpected results.  Some implementations of TTL get around this by including input pull down or pull up resistors.  However your example has neither.  

Here is a circuit similar to the one in your question:



As most logic gate transistor circuits are designed to operate in saturation, we will assume that is the case here.  One can then say the transistor is like a switch.  On when the base senses a logic 1 (high voltage) and off when the base senses a logic 0 (ground).

At this point we note that if either transistor is off the output is pulled to ground or a logic 0 by the output pull down resistor.  Only when both transistors are on do they complete the circuit to high voltage and over power the output's pull down resistor driving the output to logic 1.