For PhD acceptance rates, many schools/departments essentially require PhD candidates to find a potential supervisor prior to applying. My, non-Oxbridge, department's acceptance rate is something like 90% for students who find a supervisor prior to applying and 0% for those without a supervisor. This "requirement" really changes the dynamics of applying for a PhD.
For Masters programs, I think there are a number of minor factors and one large one that result in UK universities having a higher acceptance rate than US universities. Graduate classes/modules in the UK are only taken by Masters students. PhD students and advance undergraduates do not take graduate level classes. This means departments need to accept enough Masters students to make teaching the classes worthwhile.
Getting enough fee-paying students is hard and the competition between programs is fierce. Departments tend not to pass up on students that they think can pass. While "entry tariff" (basically the GPA of admitted students) is starting to count in league tables, in my experience UK universities are not as adapt at manipulating the league tables. Further, UK universities generally have a hard minimum "GPA" of a 2.1 degree classification in a related field. Since undergraduate education in the UK is more specialized than the US this means that a "related" field is much more narrowly defined. Minimum requirements are strictly enforced in the UK so Brits tend not to apply to positions that they are not qualified for. Overall, this cuts out the bottom of the application pool. That said, something like 80-90% of students get a 2.1, and I do not think thesee factors substantiallychange the acceptance rates.
What I think matters the most is the fact that students only apply to a select few schools. My UK students tend to apply from between 1 and 4 Masters programs. I have less experience with US students, but my intuition is they apply to a lot more schools. I think that the reduced number of UK applicants increases the UK acceptance rates.