Sandel here gets all the big things right--and a shockingly large number of the little things wrong.
His main thesis is absolutely correct: the introduction of money and markets can fundamentally change the character or nature of a particular transaction. Sandel is correct that society often does not fully appreciate this basic fact--which causes us to use monetary incentives in ways that can be more detrimental than beneficial.
Most of the time, but not all of the time. Sandel states that he's just trying to start a conversation, to get people to appreciate the non-monetary costs of monetizing a particular transaction or situation. That is a laudable goal. But his tone is universally judgmental; sometimes absurdly so. His criticism of Billy Beane's A's for making walks more prevalent literally made my jaw drop; he literally used teams playing smarter baseball as an excuse to lambaste economics.
Moreover, his criticism of economics (in particular) and social science (in general) tends towards the dogmatic. He mostly cites fifty-year-old economics texts that grandfathered rational choice theory, and holds them us as emblematic of the discipline; a discipline which has become much more nuanced over the years, in part because of the types of criticisms that Sandel gives. Economics and Rat Choice Theory do certainly have their problems, but an inability to understand why people give gifts is not one of them. (Certain rat choice models, given their assumptions, cannot take that behavior into account--but most economists would agree that this is a problem with those particular models not a problem with the gift-giver, and certainly not a problem with all of Economics.)
Finally, Sandel asserts that the general "marketization" of society is a recent phenomenon, really since the 1980s. I remain unconvinced. There are plenty of examples of these types of phenomenon before 1980--Sandel himself cites some of them--and so all we're left with is Sandel's assertions that things really have gotten worse. That's an easy, and easily believed, assertion--it's why politicians make it all the time. But asserting that things have gotten worse is a long way from demonstrating it to be true--and Sandel doesn't come close to the latter.
So to summarize: Great Thesis, Sloppy and Biased Reasoning.