Everything had gone so wrong in Sherlock's life, that he took his brother's word at face value. Plus, he might have imagined the flinching to be a result of him wanting to betray him, as well as the nerves. Mycroft doesn't look Sherlock in the eye one moment during that scene and Sherlock probably thought it was because of the betrayal, and I also imagine this is the "Mycroft is smarter and cleverer than Sherlock and can manipulate him"— plus Silas is very very carefully looking at Sherlock, if Sherlock for a moment thought that Mycroft was lying or pretending, Silas would have caught on and wouldn't have believed Mycroft because the one who knows Mycroft best at that table is Sherlock. Mycroft needed to make sure Sherlock believed him, no matter what his plan later on was, or else Silas wouldn't have believed him. Cordelia is someone I'm less willing to forgive for this, she literally speaks over/for Mycroft before he even gets to say what he wants and then continue doing that, she and Silas are both pretty toxic in different ways and both contributed to Mycroft's flinching (he flinched at the raised voices and at the fight, so that probably happened a lot in his home and it would have been Silas and Cordelia fighting) asw, but again, Sherlock believing made it easier for her (as well as Moriarty and Bea) to believe it; he it the only one Mycroft really had to convince. Plus, considering we never find out what the British govt actually tell him, but we do see that he's like ashen and afraid when he meets his family again, I think it's safe to assume that the government did tell him exactly that, which means he was telling Sherlock (it was hus belief that was important) the truth about what they wanted, and possibly what he told them because he tells them that "they'll help" and moves the convo on, so Sherlock sees the nerves, the flinching, the inability of Mycroft to look him in the eyes, and the truth of his words and comes to that conclusion— he knows Mycroft well enough to know he's telling the truth about what the govt wants and hence, scene plays out as it did. And despite loving Mycroft and not ever thinking he'd betray them (when he sees Mycroft in Paris, for example, he has no worry on why he might be there despite the fact that they just followed their Father there, and openly explains what he knew), all of that plus the fact that he'd just discovered his Father as an evil evil man, and that his sister was alive and working with him had already done a number on him.