I introduce myself and get right to the point, and always let them know I will respect their place as if it were my own, and that I do not expect something for nothing. One property I hunt, the old fellow likes deer meat so I keep his freezer packed with it. I harvest, butcher, and vac-pac a deer for him every year. One year I bought him a pressure cooker so he could make quick roasts. I also go clean up limbs & such from his yard every time there's high wind. We even go out on the bass boat time to time.
Another fellow was hesitant to allow me to hunt but didn't want to say no, either. He has a large property & already has two other guys hunting it. He is worried about safety & doesn't want his hunters overlapping, so I printed a large image of the property & outlined in red the area I was interested in, ensuring him I would not stray out of the outlined area or disturb the other hunters. With it I included my contact info and $100 gift certificate to a local eatery he mentioned liking. My taking the time to lay it all out for him plus sit with him awhile to listen to his stories made all the difference. I also promised I would mend any fence damage I come across and pick up any trash. His big concern was coyotes. He wants them dead, so this year I killed 3 coyotes and only one deer.
Another farmer I have been working on has yet to give me permission because he's paranoid about somebody shooting his cattle, plus he likes to hunt a weekend or two in gun season. To my surprise he sent word through a mutual acquaintance just a couple weeks ago that he'd like to have me come out & clear some does off his place this coming fall because they're wrecking the place and competing with his cows. I'm hoping it pans out & I can get my feet in the door because it is a huge property along a river bottom & he keeps everybody chased away.
Long story short, people are people. Land owners don't know you from Adam so it's on you to present yourself as a solid. If they're comfortable with you then you have a chance. Just be genuine and don't be afraid to grease the wheels. Chances are good they'll say no. Chances are also good they might say yes. Either way treat them with respect & dignity. I've even had one guy give me the PETA speech, lecturing me on why hunting is cruel. I let him say tell his tale, then politely thanked him for the consideration.