This is referencing the plot of the second chapter of The Adventures of Tom Sawyer, by Mark Twain. In this chapter, Tom tricks the other boys into doing his work for him:
“What do you call work?”
“Why, ain’t that work?”
Tom resumed his whitewashing, and answered carelessly:
“Well, maybe it is, and maybe it ain’t. All I know, is, it suits Tom Sawyer.”
“Oh come, now, you don’t mean to let on that you like it?”
The brush continued to move.
“Like it? Well, I don’t see why I oughtn’t to like it. Does a boy get a chance to whitewash a fence every day?”
That put the thing in a new light. Ben stopped nibbling his apple. Tom swept his brush daintily back and forth – stepped back to note the effect – added a touch here and there – criticised the effect again – Ben watching every move and getting more and more interested, more and more absorbed. Presently he said:
“Say, Tom, let me whitewash a little.”
Tom considered, was about to consent; but he altered his mind:
“No – no – I reckon it wouldn’t hardly do, Ben. You see, Aunt Polly’s awful particular about this fence – right here on the street, you know – but if it was the back fence I wouldn’t mind and she wouldn’t. Yes, she’s awful particular about this fence; it’s got to be done very careful; I reckon there ain’t one boy in a thousand, maybe two thousand, that can do it the way it’s got to be done.”
“No – is that so? Oh come, now – lemme, just try. Only just a little – I’d let you, if you was me, Tom.”
“Ben, I’d like to, honest injun; but Aunt Polly – well, Jim wanted to do it, but she wouldn’t let him; Sid wanted to do it, and she wouldn’t let Sid. Now don’t you see how I’m fixed? If you was to tackle this fence and anything was to happen to it – ”
“Oh, shucks, I’ll be just as careful. Now lemme try. Say – I’ll give you the core of my apple.”
“Well, here – No, Ben, now don’t. I’m afeard – ”
“I’ll give you all of it!”
The Adventures of Tom Sawyer, chapter 2 - taken from the PBS site
What they are saying here, is that this character is making the other people do his work for him - and that these people are doing the same thing, pushing their work onto other people. (FWIW, they are commonly referred to as help vampires across the Stack Exchange network.)
Since Tom Sawyer tricked the other people into doing the work that he was supposed to do, they are comparing these people to him.
