You'll need some overlap to transition the site, but the time left on the plan you're canceling can be credited or refunded.
WordPress is a 'server-side' application, it doesn't install on your own computer like a normal program. It is possible to create a test server environment on your own computer and run WordPress that way. Though it is moderately advanced, and the conditions are not quite the same as that of the hosting account. Here is an article on the subject, you can find more with your favorite search engine for 'install wordpress locally [windows/mac]' or something similar.