No, the statement is correct. Note that most people also apply to other good universities such as Harvard, Yale, Oxford and therefore not all people accept their Cambridge offer. This year, around 200 people are taking the LLM here (highest number ever).
It's safe to say that LSE is much easier to get into. Oxford is considerably harder, and Cambridge is known to have the highest entry requirements in the UK.
(Note that I'm just talking about entry requirements, not necessarily the quality of the programme/rankings).
I would generally second that, however this may also in which country you graduated.
Speaking for Germany, Camebridge LL.M. is the hardest to get in. But LSE is considered to be more selective than Oxford. Might be the case because Germans can only apply to the MJur and not the BCL. Camebridge apparently requires a minimum of 11.5, LSE 10.0 and rumor has it that Oxford MJur only requires 9.x.