Jumpseating is an incredible privelege. A comprehensive guide for jumpseating is provided on ALPA's website: "Jumpseat".
Each airline can be slightly different regarding how to list for the jumpseat but it all culminates at the same point when the gate agent either awards you a seat in the back, hands you a pass for the jumpseat, or informs you the jumpseat is already taken or unavailable for some other reason.
If awarded the jumpseat, it make sense why we are asking the captain for permission. We are asking to invade that crews work space, observe them, and crowd the flightdeck. We are grateful for them to invite us in. But when we are awarded a seat in the back, why does the captain care? They weren't involved in our transaction with the gate agent and the company.
One time, I was working a flight that was running late when the gate agent popped their head through the flightdeck door. In the most rushed and unintelligible speech, they said, "there are some people listed but I just want to get the flight out on time so I will tell them no okay thanks bye," and ran back up the jet bridge.
Listing for the jumpseat causes the gate agent extra paperwork. Gate agents are aggressively scored on their "on time" metrics. Some gate agents will not help jumpseaters if they are afraid it might trigger a delay with their management.
If the captain is aware of a jumpseater, the captain might refuse to depart until the gate agent does the paperwork to get that jumpseater on. Gate agents will likely be cooperative at this point, because the reason for the delay can now be coded as the captain and not themselves.
If you are new to the career, I don't mean to paint gate agents negatively, this behavior is exceedingly rare. Most gate agents go out of their way to assist. I can't help but imagine however, it's years of captains showing loyalty and integrity to complete strangers in uniform that made the jumpseating culture as good as it is today. This is the reason I "ask" captain for the jumpseat even when I have a seat in the back.
My personal script after receiving permission to the jumpseat goes something like, "I don't jumpseat often, but I am here to make friends. If I do or say anything I shouldn't or that is poor etiquette, please let me know." and it is generally well received.
If it's your first or second time jumpseating, then tell them, they will likely be more than happy to help you feel comfortable.