Maybe through translations?
"Popular" is a relative term, but Shakespeare's plays were not completely unknown outside of England during his lifetime. In The Cambridge Companion to Shakespeare on Stage professor Anthony Dawson of The University of British Columbia states that a shortened and simplified version of Romeo and Juliet was performed in Nördlingen, Germany in 1604. The show was performed by traveling English actors speaking their native language. Such performances on the continent in the 1600s were few and far between.
It wasn't until the mid-18th Century that Shakespeare began to gain a wide viewership outside of Britain. In 1771 Johann Wolfgang von Goethe organized a Shakespeare Jubilee in Frankfurt to promote the plays to the general public and translations into French, German, Italian, and Spanish began to appear.