How can I write a personal statement that doesn't sound arrogant?

This is something that everyone will struggle with: writing 4000 characters about how great you are will inevitably feel a bit awkward. The key is to be confident, but not cocky. You can do this in two ways.
Firstly, if you are writing about a particular skill or quality that you have, back it up. By justifying your statement that you have 'an aptitude for languages', for example, it's no longer a bold and overconfident claim, but a true fact about you. Provide the evidence by telling them about how you won that award from school last year, for example, or how you had the initiative to do a project of your own at home. Each characteristic, such as 'good at time management' or 'excellent at working in groups', should have a justification to it. Respectively, doing the International Baccalaureate might have demonstrated how you can juggle six subjects at once, or being a member of the football team might show your enthusiasm to collaborate with others.
Alternatively, you could show how the skills you have now aren't simply natural ability, but instead a progression over months or years, preferably which you have actively driven yourself. If possible, your personal statement should definitely not take the form of a list of achievements: universities will be more interested  in the journey you took to get there, because that will show what sort of person you are. For example, if you are now an active member of the debating society, is this something you have always enjoyed? Maybe it was the case that you used to struggle with public speaking, but you decided to push your boundaries and by forcing yourself into a temporarily uncomfortable position, you have found something that you love and are good at.