They are asking what date you want to sign a new lease til and the prices per month for signing each respective lease. But whoever wrote it is a flunkee, they will probably spend the rest of their life typing shit up at an apartment management office.
OP, it says that you can sign a new lease (more on that below) or you can let the lease lapse and go to a month-to-month system
month-to-month means you only have to live there until the end of the monthly lease, so if you decide to move in five months, you're not locked in for the rest of the lease
the upside to the lease is it looks to be about 300-400 bucks a month cheaper and you can sort of select your end date as long as it's six months out
the upside to the month-to-month is you can move out pretty much whenever the fuck you want
the checkboxes are so you can choose how long you want the new lease to go for (note the soonest you can do is july, six months out, so you can't just lease it month-to-month and get the best of both worlds)
if you do a new lease until july, for example, in june you'll probably get a letter just like this that says you can do the same thing, but the first month will be January 2017
tldr; it's a flexible leasing agreement with checkboxes for how long you want the lease to last
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