Here's how the "gift card exchange" works. You go into a Kmart holding a defunct Toys R Us gift card. Regardless of the value remaining on it (which Kmart has no way to check), a Kmart customer service desk employee is supposed to take your TRU card and give you a link to a brief survey to fill out about you and your toy-buying habits. After submitting the survey, you get a one-time $10 freecash offer added to your account, that can be spent at Kmart in-store only for about the next 2 weeks. There's no real relationship between your unspent TRU gift card and the money Kmart gives you.
The "toy of the month" thing is a $5 freecash offer, sent to eligible members' accounts once a month and expires in two weeks, during which time Kmart raises the prices of toys that had previously been on sale. The catch is that many customers who follow the instructions to send a text to sign up for this have not been sent the promised offer, with no indication as to why (they have no problems receiving other promotions).
Both programs are being advertised kinda misleadingly.