Learn English – What does “play me close” mean

Way here is used in the sense of “path”, literally, or method, figuratively; and VERB our way is a common idiom meaning to traverse that path by VERBing.

For instance, one of the most chilling moments in drama is Goneril’s line in King Lear, after she and her husband have put out Gloucester’s eyes:

Go thrust him out at gates and let him smell his way to Dover.

Almost any word can serve as VERB; here are a few examples from Google:

How German Companies Bribed Their Way to Greek Deals

Wynonna and partner Tony Dovolani—who was Sara Evans' partner years ago—cha-cha-cha’ed their way to an 18 out of 30. (The cha-cha-cha is a Latin dance)

The team that could build the “coolest” machine out of a pile of scrap metal and spare parts would receive a $10,000 prize. In the meantime, the festival-goers could look on as the hackers DIY-ed their way to steampunk Rube Goldberg bliss. (DIY is do-it-yourself)

‘Did the Patriots cheat their way to another Superbowl? … ‘No, they Tom Brady-ed their way to another one.’ (The New England Patriots are a football team, and Tom Brady is their quarterback)

Ron is saying "To get across the room we are going to have to play chess."