Authors using anonymous publication was actually very common in the 17th century. The reasons that authors had for this practice were presumably varied and, to historians, not always clear.

Propriety could have been a factor. For example Hobbes published "Elements of Law" (which had a target audience among the gentry) anonymously, possibly as he was a commoner and wished to avoid embarrassing his patron. Spinoza's more inflammatory and deeply political "Tractatus theologico-politicus" was probably published anonymously to make him less of a target for the Orangists.

However not all reasons are as clear as Hobbes and Spinoza's. Locke published defenses of the Glorious Revolution anonymously long after James II was chased from England. Maintaining this anonymity his whole life before revealing it in his will. Newton is another case of unclear anonymity. He continued to release anonymous publications (and not particularly controversial ones) deep into his career when he was a powerful courtier and the most respected scholar in the western world. Caution also does not seem to fit with what else we know of Newton.

Which brings us to Descartes. His only anonymous publication "The Discourse on the Method" wasn't a particularly large secret. It is full of autobiographical detail, was widely attributed to Descartes by Mersenne and, when republished in Latin 7 years later, it had Descartes name on it. The reasoning for him publishing this treatise anonymously is unclear but may have been due to Descartes viewing himself as being in an academic war with the Jesuits. Perhaps by maintaining (at least originally) some anonymity he hoped to immunise himself from criticism. Descartes reserved his most biting criticism for the scholastic class and the universities who, in Catholic Europe, were almost synonymous with the Jesuits.

You've said in your question that a regular Joe couldn't have understood the "Discourse on the Method" but along with the "Meditations on First Philosophy" that's exactly who his target audience was. These two publications were published in French while Latin remained the language of academics. He had hoped to appeal to readers who had no background in philosophy. The educated philosophers remained, in his mind, his foremost intellectual enemies. Anonymous publishers notes (widely attributed to Descartes) at the front of many of his publications offered scathing criticism for academics in particular scholastics. While in his personal correspondence to Mersenne, Descartes complained of the pressure to self-censor "Le Monde" for fear of offending Aristotelians within the ecclesiastical hierarchy. In this correspondence we see Descartes specifically reference the Galileo affair.

It's also worth pointing out that the Church and its members was a many headed beast without monolithic opinion. The academic period in question is often referred to as second scholasticism. This period was distinct from earlier medieval scholasticism for its scientific progression and its many schools of thought. Although Aristotle and St Thomas remained the foremost philosophical influences. The church in this period was widely able entertain non-orthodox ideas, including during the early stages of the Galileo affair. There remains a debate as to how much of the escalation of the Galileo affair was academic and how much was political (but that's a whole other and much longer post). Regardless as previously mentioned Descartes, in at least one publication, had felt the need to self-censor his criticism of Aristotelian physics because of the affair.

It is unclear that Descartes work was better received amongst the general populace. But it is probable he directed two of his major publications towards them hoping to find a receptive audience. It is also possible he was better received among the less educated as Cartesianism was not particularly opposed to Catholicism, which had a buy in amongst academics and the general populace. It was however contrary to Aristotlean physics, which had a buy in amongst most scholastic academics but remained largely unknown to the general populace.