This is a great question, as it allows us to consider the way people talk about artists' and architects' contributions via shorthand and also to consider the different things meant by terms like "antiquity" and "classical".
To start, I wouldn't characterize Brunelleschi's contribution as "reintroducing" the classical orders of architecture, as it's not too difficult to find earlier examples of people using orders that we consider classical—e.g., the capitals on Nicola Pisano's Pisa Baptistery Pulpit, from about 1260 are Corinthian. Nicola is even using what we might term a "classicizing" rounded arch (albeit a trilobed arch, which is perhaps decidedly un-classical to our eyes). Depending on how you define "classical" you could probably find earlier examples of artists and architects "reviving" ancient forms (e.g., Carolingian art). That said, the phrase is often used as convenient shorthand for what makes Brunelleschi noteworthy.
Brunelleschi does make contributions that are directly pulled from his interest in and study of classical art and architecture. The most noteworthy example is his dome for Florence's Cathedral, which is rightly considered a marvel of engineering—and which can be favorably compared with ancient structures. The dome was the largest since the Pantheon in Rome, measuring about 138' across and 180' high. Building a dome on this scale required rethinking basic principles of construction: the usual method of centering, in which arches and domes were built on a wooden framework, wouldn't work here, because (1) there was no timber large enough to span this space/distance, and (2) the wooden centering would have been too heavy to support itself and the dome. Brunelleschi developed a system of internal scaffolding that supported itself and had floors that could be lifted as the work progressed. He also invented machines to hoist materials, supplies, etc. All of these innovations are likely due to his studying ancient structures and trying to understand how they were constructed. The brickwork (a herringbone pattern) of part of the dome also calls to mind some of the brickwork of antiquity (which he may have seen on trips to Rome). In these and other design principles we may think of Brunelleschi as someone calling on "ancient" traditions or tactics, even if he is not directly copying antiquity in its specifics.
The lantern of the dome (completed by Michelozzo after Brunelleschi's death but per his design) shows some of his classical interest, as well. The clamshell niches, for example, or the volutes/scrolls, are notably classical, as are the Corinthian capitals on the pilasters.
More broadly, Brunelleschi was an architect with a consistent interest in antiquity: he went to Rome on several occasions, he studied ancient ruins, and he adapted what he saw to modern needs. But it's also important to note that for Brunelleschi (and others) "antiquity" could mean different structures than what we think of today as ancient. He pulled from Romanesque structures, e.g., which in some cases might have been considered ancient. The most famous example of this is the Florence Baptistry, which Florentines believed to be an ancient temple originally dedicated to Mars but which was actually built and consecrated in the 11th century (NB: there is some debate on how much Florentines really believed this and how much they just chose to ignore the facts; I can't find the article on this at the moment, but if/when I do I'll add it here). I'm oversimplifying here, but "antiquity" can be thought of as anti-Gothic, and was not necessarily limited to works that fit within a strict date range of the Roman Empire, e.g. (they didn't have our methods of dating, remember).
Art historians, when talking about Brunelleschi's classicism, also often invoke more abstract concepts: a certain harmony to his forms, mathematical ratios and proportions that govern his designs, an interest in perspective, and so on.
If anything, it's Brunelleschi's coherent and more global incorporation of what he learned from studying antiquity that is noteworthy. Lots of artists and architects could and did drop ancient quotations (classical capitals, columns, statuary, etc.) into their works, but Brunelleschi went beyond this to meaningfully synthesize what he learned from antiquity into his own work.
The TL;DR of his classicism might be summed up in this way: Brunelleschi was an architect who took a particular interest in applying what he learned from studying Roman ruins and buildings (and those believed to be Roman) and applying them to modern problems of design and construction in a way that others were not yet doing.
Ross King's book Brunelleschi's Dome is an adequate popular book on Brunelleschi, but for something a bit more scholarly you might check out the following:
As an aside, the Florence Dome presented a major problem for contemporary architects, and it seems that construction on the cathedral was underway even though no one in the 14th century had a clear sense of how exactly to build a dome this size. According to Giorgio Vasari (whose accounts must always be taken with a huge dose of salt, if not an entire salt mine), one idea proposed was the following: "[there were] men who said that it would have been a good thing to fill it [the cathedral] with earth mingled with small coins, to the end that, when it [the dome] had been raised, anyone who wanted some of that earth might be given leave to go and fetch it, and thus the people would carry it away in a moment without any expense. Filippo alone said that it could be raised without so much woodwork, without piers, without earth, without so great expenditure on so many arches, and very easily without any framework.” Vasari, Life of Brunelleschi
Yes, apparently one idea (per Vasari) was to fill the whole church with dirt in which money was hidden, build the dome on the dirt, and then have people excavate the dirt as they searched for the coins. It was a special time.
