Leech’s joke here is the smugness of the father of the family (paterfamilias), who would rather inconvenience his family by camping on the beach instead of staying at a hotel. This large family putting up with the man is an implicit joke because no one protests or scowls. The man peeling potatoes (left foreground) probably alludes to Charles Dickens, Leech’s friend, fond of boiling potatoes at picnics. You’ve got to smile (at least) at the notion of Dickens demanding potatoes for his picnic; “On Tuesday, we are going on another picnic,” he wrote to a friend, “with the materials for a fire, at my express stipulation; and a great iron pot to boil potatoes in. These things, and the eatables, go on in a cart.” [Elsewhere, the potatoes blight was wrecking Ireland’s health and economy. But on the Isle of Wight, there was no scarcity.] The rest of the food being prepared is too indistinct to identify.
See John Leech. Paterfamilias Makes Himself Independent of Hotels (1854), Punch, or the London Charivari, vol. 26. London, 1854
http://books.google.com/books?id=LVMPAQAAIAAJ&pg=PP14&lpg=PP14&dq=Punch+1854+paterfamilias+makes+himself+independent&source=bl&ots=DZQq5YIQKW&sig=AtS_7wI0ZHMd3ub1S5_HshBejMc&hl=en&sa=X&ei=s8jFUsqeEaXmsASr9IDwDA&ved=0CDIQ6AEwAQ#v=onepage&q=makes%20himself%20&f=false;Letters of Charles Dickens 1847-1849. Graham Storey and Kenneth Fielding, eds. vol.5 Clarendon Press 1981, 588.