This crux of the deal for both teams is essentially this:
To Toronto: More contract flexibility (Leafs were up against the 50-contract limit, & this clears four spots)
To New York: $5M in freed-up cash, $3M in freed-up cap space
Since Grabner's on an expiring contract, the Leafs may also be hoping they can flip him for a pick at the deadline if they rehabilitate his trade value.
As for the players you're getting, Nashville fans could probably tell you more about Beck (we acquired him earlier this summer), but he looks to be a decent depth 4th liner at the NHL level.
Nilsson & Gibson probably have the best chance of getting to the NHL out of the prospects you got. Gibson's only 22, and ended up winning the starting job for our AHL team last year by the end of the year, while putting up pretty good numbers. Nilsson is a prototypical "stay-at-home" defenseman, though he doesn't have a ton of size.
As for Finn, he's a prospect that a lot of Leaf fans were high on. In his last year of junior, he captained the Guelph Storm of the WHL to the Memorial Cup tournament, and put up good offensive numbers. But he had a really awful 1st year in professional hockey last year where he couldn't stick on the AHL roster, and got demoted to the ECHL. And it's really looking like his skating may just not be good enough for professional hockey.
Lastly, there's Carter Verhaeghe. He had a good offensive numbers the year after we drafted him, and essentially duplicated those numbers again last year. But he's slipped down our prospect chart at center, and probably would have been squeezed off our AHL roster and sent down to the ECHL, or relegated to a 4th line role if he'd remained in the organization. It's also worrisome how much his offensive production in junior seemed to be tied to having Brendan Perlini as a linemate. His offensive rise the year after he was drafted coincided with Perlini's, and then when Perlini was out with an injury for the first few months of last year, Verhaeghe's numbers fell to below a point-a-game, and didn't rebound until Perlini returned to the lineup.