Confused with PC RS232 pin-out for this stepper motor communication
On the 9-pin D-type of a PC's RS-232 port, the signals are

 1 DCD in    data carrier detect
 2 RXD in    received data
 3 TXD out   transmitted data
 4 DTR out   data terminal ready
 5 GND =
 6 DSR in    data set ready
 7 RTS out   request to send
 8 CTS in    clear to send
 9 RNG in    ring indicator


By joining 1, 6, 7, 8 they are telling you to connect PC's output RTS to the inputs CTS, DSR and DCD.  Although I have no knowledge of your receiving device, this is a typical method to convince software (on the PC) that there is a far device asserting DSR, DCD and CTS.  It means the output RTS (I am ready) is sent back as "other side is ready".  (I'm skirting over the distinctions between DSR, DCD and CTS, but some software will require all three to be asserted before sending data out of TXD.)

The purpose will be solely to make whatever software is running on the PC actually transmit characters.  If your software will transmit characters without needing this, you won't need this bridging.  With many RS-232 related topics, the best thing to do is try your software and see if it communicates.  If you don't have a far end you trust, try a scope, LED or fast voltmeter.

PS.  The bridging of pins 3 and 7 on the servo side are unique to this device, not standard RS-232 wiring at all.  See the note in the referenced manual about TX Pulldown (pin 7), required to be connected to TXD (pin 3) "if addressing is not used", which is something specific to this device.