Hello! I go to a grammar school for my sixth form, and I used to go to an ordinary, kinda crappy school, so I am in the same position you'll be.
Well, I'd say it's not because of tutors. From what I see, the teachers actually love teaching here. At my old school I never got that impression. To join the grammar school in the first place, students normally have to have some sort of proof that they are smart (the school I go to makes students get a certain score on the 11+ test). So it means that for starters, the students are somewhat smart.
But I'd really say it's just the teachers, because they limit who can get in, the classes are quite small. At the school I'm at there are 90 students per year in the lower school (250 per year in sixth form), which means that their learning can be a lot more, like, specific to just them. The teachers seem really really good though, and there are less behaviour issues because in general the students want to be there, so classes get disrupted a lot less, so students and teachers can both work to their full ability, ensuring high marks for the students.
I think the difference in the style of teaching is quite great, but that's more because you're switching to A levels, rather than going to a grammar school. The classes are smaller than GCSE ones, and everything kind of goes more quickly, but the teachers all seem really smart, and you kind of get the impression they've taught the same thing over and over so they tend to know exactly how to get the marks in the exam. I'm absolutely loving the grammar school though, and I'm sure you will too, it's not as... 'snobby' as I thought it would be, fortunately, if that makes sense? But it's still very academic, the students here are overall quite nerdy and I love it XD.
Anywho, as for help with the application, do you know what it consists of? For us we just had to write our contact details and what A levels we wanted to do, but that may be different for other schools. We then had to have an interview with someone high up in the sixth form teaching, and then we were just asked things like why we want to go to this school, you may be asked about your subjects, or if you have any questions about the school. Sorry if I'm not being much help, if you have to write to the school or something, I'd say just seem interested and make sure they know that you want to go there. I don't know how hard it is to get in to your grammar school sixth form though.