What can be the reason for the extra connections from this Wheatstone bridge to the amplifier?
The extra +/- excitation terminal are for a Voltage Sense.  When transducer cables are long, there could be IR voltage drops along the cable, messing around with your transducer sensitivity, or passing noise along.  The sense lines provide a LOW CURRENT sense path (assuming they're connected to high Z inputs), telling a power control mechanism what the excitation is at the bridge, facilitating feedback control of the excitation.