“The last year” is the time from 365 days¹ ago to now. As TRomano explains, this is different from “last year”, which is 2015. So an action that happened over “the last year” is either still ongoing or has just finished.
In the first part of the sentence, both the present perfect and the simple past are possible:
Happy Meals have spent the last year touring the globe.
Happy Meals spent the last year touring the globe.
The first variant implies that they're still touring the globe, or maybe they've just come home but there is still a sense of immediacy, the fact that they're touring is still current or recent. The second variant implies that the touring is over, they are now back home.
In the sentence you quote, the present perfect is used because the sentence is dealing with a consequence of the touring. The band have been touring and they have been experimenting with sound and these experiments have helped them produce their new album.
On the other hand, the simple past would be used to indicate that the action is finished, for example
Happy Meals spent the last year touring the globe, but now they are starting a series of concerts in their home town.
¹ or 366 days, or approximately one year.