its badly explained, but what the first paragraph is saying is basically:

"the act of measuring time in greater detail requires increasing amounts of energy" (increasing amounts of energy dilates time because general relativity) "therefore, time-keeping devices become less precise".

time dilation implies "less precise measurements". If we assume a balance where time measurements always require more energy, then there has to be a level of precision at which increasing precision through deploying more energy increases precision by less than the increased energy decreases precision, therefore, improving precision becomes impossible.