Ah, that explains a few things! Indeed you have a number of problems and as it stands you are likely to blow up the Arduino. Your problem with the batteries is a Fritzing quirk, the bendable leads on the battery are both female and thus won’t connect to each other. You need to run the wires from the positive of the 9V battery to an empty row on the breadboard and then connect the negative wire from the 3v battery to the same row on the breadboard to actually get a connection (this is why schematic shows the batteries to be unconnected to each other). Your current connection would work in real life but not of Fritzing. Alternately you could use only the 9V battery and note that it is in real life a 12V battery. Now the negative side of the battery wants to go to ground and the positive side of the battery wants to go to Vin on the arduino (I recall checking the schematic for the arduino once and deciding that you could power Vin from there if you didn’t use the USB connector or the barrel jack on the Arduino but you may want to verify that). The Vin connection (expecting 9 to 12V in) feeds voltage regulators on the Arduino that provide 5V and 3.3V out to the pins labeled that. You are attempting to put 12V on the 5V output to the Arduino which is likely t blow the board. If your motors want to run from 12V (which is likely, the Arduino regulator can only supply a limited amount of current at 5V and likely won’t run the motors successfully) they too would need to connect to the battery positive or the Vin pin. Motor U4 is connected wrong. It should connect to the motor output on the LM293 not the IO pin on the arduino (which doen’t have the capacity to drive a motor and would probably damage the Arduino). I’m unclear what the transistors and diodes are supposed to do, but I don’t think it is correct. As noted before the motor driver probably has appropriate back emf protection but you would need to verify that (a quick look at the data sheet indicates that is correct). In your current configuration the motors will only go one direction (which may be what you want), to go both directions you would need to use two drivers per motor (i.e. one l293d will only drive 2 motors bidirectionally). I don’t think you need either the transistors or the diodes. Hope this helps some.
Peter