By listening to recordings. 'Jazz Standards' can refer to both songs from the American Songbook (Gershwin, Cole Porter, etc.) as well as songs written by jazz musicians (Ellington, Monk, Bill Evans, Joe Henderson, etc.).
For the former, the original musicals or movies these songs come from are a great place to start. Then find jazz singers you like. I would recommend Ella Fitzgerald. For the latter, find tunes you want to learn and musicians you gravitate towards and check out the recordings.
As far as actually learning the tunes, the best way is to actually transcribe them yourself, lyrics, melody and chords. In addition to developing your ear to hear melodic and harmonic relationships (which is important if you want to improvise), transcribing from recordings will also help you learn the phrasing used by singers, as opposed to just the quarter-note rhythms you might see in sheet music. Fake books also can have errors.
Then you need to play with other musicians.