He didn't "make up" that many just from scratch, he modified existing words or adding different endings or changed them slightly. For example, the most famous word seems to be "elbow." But shakespeare didn't invent the word for that body part, he just used it to describe poking someone with that body part.

Modified words when used in context aren't too difficult to figure out, especially when used humorously or with just a different suffix/prefix to change the meaning. An extreme example of this is James Joyce's Ulysses, and of course Finnegan's Wake. While the average reader may have trouble with these, keep in mind that many of the words in Ulysses, and most of the words in Wake, are "made up," while Shakespeare would used a made-up word very rarely in a clear context.

I would ask an additional question :

How do we know he actually invented any words whatsoever?

Culture needs a witness and some form of surviving account for us to recognize it. Shakespeare's works have survived to this day because people in every age have deemed it worthy to pass on. But the vast majority of writing from his time has simply disappeared. So some of the words that Shakespeare used is the earliest surviving example of that word that we have.

But That certainly does not prove that he invented that word.