OK. So you want to duplicate that circuit (actually two circuits, and with six digits instead of four), using larger displays that contain several LED elements in series and need 12~15V across the segment. Right?
You need another supply rail at a higher voltage, and you need level shifting circuitry.
Your higher-voltage supply would normally be a positive rail that's a few volts higher than the voltage drop of a segment. In this case I'd recommend 19V or more. Laptop power supplies are typically rated around this voltage and might be a convenient power source. You can derive the 5V rail from the higher rail using a linear or switching regulator.
Voltage shifting is normally done with a 0V-referenced common emitter switching transistor (or MOSFET) (NPN or N-channel) driving a second common-emitter switching transistor (or MOSFET) (PNP or P-channel) that operates from the high voltage supply. When the control signal goes high (+5V), the first transistor turns ON and provides bias to the second transistor, which turns ON and provides the positive supply rail to the display.
I think you should take this opportunity to reconsider the interface between the micro and the display. Specifically, the external BCD-to-7-segment decoder causes two main disadvantages: it limits the characters you can display, and its outputs use TTL/LS levels, which are more difficult to deal with.
I would reserve an 8-bit port to drive all of the individual segments, including the decimal points, independently. This will require more I/O but I think it will be worthwhile, because you can display arbitrary symbols such as dashes and some capital letters.
You will need two of these ports; one for each six-digit display. The other option is to multiplex by 12 instead of by six, but I wouldn't do that; the peak current and the electrical noise would be doubled.
You can however use an external selector to reduce the number of control signals needed for the digit selection. The commonest choice is the 74xx138. In this case I would use a 74HC138. There is also a 74HC238 which has active-high select outputs, which may be more appropriate, depending on the interface circuitry. (The 74xx138 has active-low select outputs.) Using a 3-to-8 decoder/selector reduces the number of I/O lines required to select the digit from six to three.
So with those changes you would need the following I/O:
3 outputs to drive the 3-to-8 decoder for digit selection;
8 outputs to drive the segments and decimal point for the first display;
8 outputs to drive the segments and decimal point for the second display;
possibly, extra outputs to drive other LEDs such as colon indicators between pairs of digits, if you want that feature.
The last factor that determines the best interface circuit to use is the arrangement of the displays - common anode, or common cathode. Either can be used; the drive circuitry just needs to be different in each case.
Assuming you can use either common anode or common cathode displays, and assuming that you will use separate 8-bit ports to drive the segment, my recommendation would be:
1. Common anode displays.
2. Both displays multiplexed together, i.e. a single digit selection control and two independent sets of segment drives.
3. Three-bit digit selection using a 74HC138
4. N-channel MOSFETs (6) in common-gate configuration with sources driven from the HC138 outputs and gates tied to +5V, with drains driving the gates of P-channel MOSFETs (6) with their sources connected to the high positive supply and their drains driving the common anodes.
5. Two sets of eight N-channel MOSFETs (16 total) with their gates driven from two 8-bit I/O ports on the microcontroller and their sources grounded, with their drains driving the cathodes of the segments through individual current limiting resistors (16).
Have a good think about what you want and let me know; I'll draw up a schematic.
Edit 1: Also, can you use SMT devices in this project or do you want to use only through-hole?
Edit 2: Do you have any particular display devices in mind? If so, can you give the part number and a link to the data sheet?