What's the meaning of "Gets me every time"?

Solution 1:

The phrase means that whatever it is engenders some strong emotion: here it will be laughter, but in other contexts it might be tears, surprise, or embarrassment.

Solution 2:

"(something) gets me" is a catch phrase, everytime is an intensifier that suggests 'I can never get used to it'.

Eric Partridge in A Dictionary of Catch Phrases explains that gets me!

that gets me! is recorded in HLM, 1922, as a 'picturesque' phrase. Meaning: 'That annoys me': recorded by the DAE for 1867; current in Britain c. 1919-39. P.B.: but still current, in UK, by allusion, in remarks like 'What gets me about it is the way she can so blatantly...'
A.B., 1979, amplifies: 'Sometimes, that really gets my goat! [also Brit.] and [the later] that really does piss me off, which is often truncated to I'm really P.O.'d!... I don't think the "got" one is largely currently, but it isn't dead. ...

Loosely applied, the phrase could also mean never fails to amuse me or such.