The Halkett boat was the brain-child of Peter Halkett, a lieutenant in the Royal Navy. It was singular not only for its use of waterproofed rubberized cloth to make an easily portable inflatable boat, but for design features which rendered it especially valuable for use on expeditions where both land and water travel might be rendered necessary by conditions such as drifting ice, open leads in the water, or difficult portages. He even developed a version of the boat which could be disassembled and worn as clothing -- this is the type shown here. The sail doubled as an umbrella, and the oars as walking sticks; the boat itself could be disassembled and "worn" as waterproof clothing by two men.
The design earned early praise from Sir John Richardson, who was likely the one who recommended it for Franklin's use. The design had only just been finalized, with Halkett having tested it on the Thames in 1844. The boat brought by Franklin was not, technically speaking, an officially-supplied item, but Halkett was eager to hear how it might perform in the Arctic, and Franklin was willing to give it a try.
Remarkably, there is a body of Inuit testimony which confirms the use of a Halkett boat; their description of it is accurate enough that there can be little doubt they saw one in use:
"[Aglooka had with him] a boat that had places on the sides that would hold wind ... with hollow places in the sides for wind (air) to hold it up when in the water ... There were sticks or holes for this boat, to keep it open (spread) when needed. This small boat was wrapped or rolled up in a bundle or pack, and carried on the shoulder of one of his men" (qtd. in Woodman, Unravelling the Franklin Mystery, p.309).
Unfortunately, since John Rae's and James Anderson's parties were also equipped with Halkett boats, and Rae was also known as "Aglooka," it's very difficult to determine with certainty whether or not this Inuit account refers to Franklin's men. It's widely assumed that some of those who crossed Simpson's strait did so in a Halkett boat, though accounts of an overturned whaleboat on the mainland side suggest this is not the only viable explanation.
But for more on this remarkable invention, I can only recommend the Wikipedia entry, which is far more detailed and better illustrated than anything I have seen in printed sources.