The title is a play on A Streetcar Named Desire, a play later made into a film.
Tweety stumbles into Sylvester's house looking for shelter and Sylvester hesitates if he saw a tweety bird in the same manner Tweety wonders if he saw a 'Putty Cat'. Sylvester snatches him inside but has to hide Tweety in a vase covered by books, when Granny appears. While an injured Hector remains bedridden, Sylvester causes whatever diversion he can to stop Granny from spotting Tweety, making Granny give multiple doses of medicine to Hector. When Hector gets in Sylvester's way from eating Tweety, Sylvester injures himself. Tweety spikes Hector's medicine resulting in Sylvester ingesting the disgusting stuff.
On ABC, the end where Sylvester is laid up with injuries had the scene of Hector bashing Sylvester's broken foot replaced with Tweety mixing additional medicines in Sylvester's bottle (which also had to be shortened due to fears of imitability).
The Looney Tunes on Nickelodeon version left in Hector bashing Sylvester's broken foot and Tweety mixing in the medicines in Sylvester's bottle, but cut off Tweety's final line, "That puddy tat's gonna be in an awful pwedicament when that medicine starts to work" and ended the cartoon with Tweety laughing as Sylvester gags and spits out the medicine (possibly to avoid implications that Tweety may have poisoned Sylvester).
When Granny returns to her knitting, there is a split second where her entire torso area disappears.