If people today have a hard time understanding what’s going on in his plays was it the same situation back in the 1500’s for the normal people who watched it?
Yes, they were. The biggest obstacles to a modern reader's comprehension of Shakespeare's plays in performance are their language and their cultural context -- two things that contemporary audience members, regardless of their social station, would have been soaking in all their lives. English Renaissance drama often dealt with political and historical themes that were well within cultural memory for English audiences; the core players might be well familiar from earlier plays, songs, and written works featuring a certain king or a certain folk hero. Plays could be written on themes as ancient and closely linked into national mythology as King Arthur, or as ripped-from-the-headlines as a real-life murder. (Check out A Yorkshire Tragedy.) Plots were selected for their relevance and their dramatic potential; they weren't received at a distance from some higher literary authority. Even plays written in deliberately-complicated verse were performed in the same vocabulary that audience members would hear at home and at work; a modern reader might need a primer to get into the nuances of "you" and "thou", but there wasn't a single member of the Elizabethan audience who didn't understand what it meant when a character in a moment of hot temper switches from consistently using "you" to impetuously using "thou" when addressing their social superior. These conventions of language were part of their milieu, just as the conventions of the stage were.
Elizabethan audiences knew Elizabethan language. With perhaps a few exceptions, they knew what bawds were; they knew what it meant to sup and wot, to commend someone to someone else, to prate and rail. They detected dirty jokes, swear words, and literary allusions on their own day-to-day instinct, rather than needing a director or a high school teacher to explain the four humours or why someone might say "zounds". They reflexively understood euphemisms and even bungled idioms because those were drawn from then-contemporary language and figures of speech -- in order for you to understand Mistress Quickly's exhortation that the dead Falstaff is "in Arthur's bosom" and to find it remotely funny, or even to clock that it's supposed to be a joke, you have to be familiar with the idea of a deceased person resting "in the bosom of Abraham". Take away that contemporary religious context or that particular Scriptural flourish and the scene makes less sense. This wasn't language from a distance, the kind of language you put on when you're pretending to read from the King James Bible for comedic effect; it was quite immediate. Shakespeare's audiences saw Shakespeare's plays performed by people who never had to consult a dictionary or a footnote to understand the words they were saying. (If you've ever seen a Shakespeare play or any other play performed by actors who clearly didn't understand what their characters meant, you know what a difference that makes.) The presence of multi-part history plays in Shakespeare's repertory -- plays that rely on you remembering who characters are and what they've been doing across multiple installments -- suggests audiences processed and understood the plots and characters they saw enough to come back for multiple sets of sequels.
Shakespeare's plays are not representative of absolutely all plays performed on stage over the course of his career -- Shakespeare had dozens and dozens of competing contemporaries whose works were being performed at the same time, and it was in the best interest of an English playing company to manage the material in their repertory of performances to conform with contemporary tastes and trends. The ultimate proof that people were able to understand Shakespeare's plays is that they kept going to them. As a writer and as an actor-turned-shareholder, Shakespeare had a strong interest in the financial well-being of the theaters at which his plays were performed and the companies of actors who performed them. If he wanted to crank out rarefied intellectual exercises that only delighted his fellow literati, he would have done so, rather than taking up a substantial financial stake in the Lord Chamberlain's Men. The patronage of aristocrats alone wouldn't sustain a Bankside playhouse -- if you wanted an audience of aristocrats, write and perform your works at court. Nor does it seem likely that aiming directly at the lowest rung of the status hierarchy would necessarily make a playhouse and its shareholders rich; the very very poor were unlikely to have much in the way of both spare money and leisure time. To make a profit at the Swan or the Globe, you also needed to appeal to the middling type of Londoners, urban working people at various levels of prestige and financial security -- craftsmen and craftswomen, merchants, families and children, young apprentices, all at various levels of formal education. All these normal non-aristocratic people possessed varying tastes regarding comedy and tragedy as well as varying levels of viewing comprehension -- and all these people attended the Elizabethan and Jacobean theater. The manner of the average person's attendance was dictated by their leisure time and financial state, the weather, public health hazards, public censure, and politics, but it was sufficiently frequent to support numerous playhouses and to bankroll the writing and performance of hundreds of plays besides Shakespeare's own. These playhouses were plonked down alongside much less intellectually exhausting venues in which to spend your Saturday off work, like animal-baiting houses and brothels -- go bet on sports, go get laid! -- and yet they were so profitable that they supported a host of other industries, from the sale of beer to thirsty audience members to the secondhand purchase and modification of cast-off clothing to furnish the playing companies with authentically luxurious costumes. It's impossible to imagine audience members paying up en masse in order to passively receive entertainments which had no entertainment value, which they did not understand, and which had very little of the abstract artistic cachet we now associate with Attending The Theatre. Would you spend a chunk of your weekly paycheck to go to the movies if you didn't understand what was happening and didn't like what you saw?
Shakespeare's audiences had some degree of access to both the writers of their favorite plays and to the actors who played their favorite parts. Industry types were known to frequent particular establishments, like the Mermaid Inn -- locations which made a tidy haunt for writers to carouse and confer among their peers as well as for fans and critics who weren't themselves men and women of letters to give their feedback on recent creative endeavors. Writers for the playhouses were more like local celebrities than distant idols. This accessibility doesn't guarantee that all authors were cheerfully receptive to criticism (have they ever been?) or that all audience members were lining up to give their input on what they'd just seen for a couple pennies, but it made the barrier between audiences and authors a lot more permeable than that between a modern audience member and a long-dead playwright who lived centuries and continents apart and spoke from a significantly different lexicon of language. The one thing Shakespeare's audiences definitively lacked is the benefit of hundreds of years of scholarship on his works -- they couldn't step out and buy a nicely bound copy of Shakespeare's complete collected works in order to follow along. No audience member ever achieves a total understanding of what they see in a Shakespeare play -- that complexity is what gives Shakespeare's plays and those of several of his contemporaries their lasting appeal even hundreds of years later. It's that complexity in imagery, plot, and characterization that leads to rewatch value, even for those who do intuitively grasp the language of what they're reading or viewing.
Some reading: