Help "emulating" a keypad dailer with a landline dialer using Arduino

Hi, I have a curious project. I'm making a kiosk which dials over a landline. The user chooses a property which they'd like to dial and the software is to interface with a hardware dialer using the Arduino to make the call over a landline.

So far I have the hook switch connected to the Arduino so my app knows the status of the receiver (on/off). The Arduino is also relaying the hook switch status to the dialer board. This is illustrated with the green rectangle in the attached image.

Here is my challenge. The board is designed to be connected to a physical pin pad, over 7 pins. It uses the rows/columns method to dial a number. E.G. -- to dial "1" you have to use pin 1 (row 1) and pin 4 (column 1). To dial "5" you'd use pin 2 (column 2) and pin 5 (column 2).

I need to be able to have the Arduino connected to the 7 pins and "write" to the pins, based on the logic that I've described above. I got the software part but don't know how to continue on the hardware part.

Help please!