I think the joke is a play on words.  'Weight' sounds like 'wait', and the phrase 'can't stand the wait' typically means 'is having a hard time remaining patient'.  Since the speaker is talking about a diet plan and being healthier, the word 'wait' is replaced with 'weight' meaning body weight.
One gruelingly intense month into the year - long diet regimen , the overwhelmed patient concluded she could not stand the weight .