Filippo Brunelleschi didn't start the renaissance anymore than The Beatles invented Pop Music. The Beatles were inspired and built upon the immensely popular work of Elvis Presley, who in turn built his own sound on a variation of upbeat Rockabilly and African-American Rythm-and-Blues. Other bands in the 1960s like The Who and The Rolling Stones also had their unique and influential place in the pantheon of Rock Music, but it didn't stop there: tons of bands in the 1960s who were less enduringly successful nonetheless influenced later musicians in myriad, perhaps more subtle ways; from San Francisco based Its A Beautiful Day all the way down to The Monks, a short-lived but influential avant-garde band founded by American GIs stationed in Germany.
Just like the Beatles were writing their music within the broader context of the 1960s and its evolving culture, there were hundreds, or actually thousands, of artists active in Italy at the time Brunelleschi was active. Indeed, in the 14th century there emerged a number of architects who embraced a more artistic approach to construction, which previously had been seen as a purely technical and mathematical affair. Jacopo della Quercia and Lorenzo Ghiberti, for example, were two artists whose contemporary reputation was almost on par with Brunelleschi's. But good old Filippo managed to get his hands on more important commissions and brought his artistic eye and training to architecture on a much grander scale than anyone else. That, in part, is why we remember him today. Would buildings be still constructed exclusively by engineers if Brunelleschi hadn't put down his chisel and picked up a drafting board? Probably not; but I'll do my best to explain why he was able to accelerate the trend of more "artistic" architecture through his own work, just like The Beatles were most prominent vehicle through which the sound of "Guitar Bands" of the 1960s developed.
Brunelleschi had a number of personal advantages which allowed him to complete the transition from sculpture to architecture. For starters, Brunelleschi came from a well-off background. His contemporary and rival Ghiberti was the lovechild of a goldsmith and the dissatisfied wife of the fourteenth-century trust fund kid, while the also very famous Jacopo della Quercia was the son of an itinerant woodcutter. On the other hand, Brunelleschi's father was a successful notary who had even been tapped by the Florentine government to support diplomatic missions. While most sculptors who were interested in architecture could only hope to self-teach themselves if they successfully won increasingly large commissions incorporating more architectural responsibilities, Filippo received a classical liberal arts education and crucially was also taught mathematics and geometry at a young age.
Rather than be apprenticed to an artistic workshop, where he would be expected to rapidly learn to earn his own keep, the Brunelleschi household was able to hire tutors to teach Filippo paint and sketch. His father, recognizing his talent and interest in the arts, arranged an apprenticeship in the workshop of a family friend when Filippo was already a few years older than the usual age at which boys were apprenticed.
Brunelleschi started earning his first commissions in his early twenties, getting subcontracted to provide small bronze sculptures for projects in the Florentine suburbs. Undoubtably he had talent and soon started earning commissions in the city of Florence proper, even developing a tense rivalry with the aforementioned Lorenzo Ghiberti.
Brunelleschi's definitive shift towards architecture seems to have occurred between 1402 and 1404, when after losing a prestigious commission to Ghiberti he relocated to Rome in a fit of frustration. Accompanied by his teenage apprentice Donatello, Brunelleschi sketched and studied ancient roman ruins extensively while producing small sculptures to maintain himself. Developing a deep love for the city of Rome, Brunelleschi would return for multiple short trips over his lifetime to continue studying ancient architecture.
On returning to Florence, although he initially returned to sculpting, Brunelleschi also dedicated himself to perspective drawings, sketching detailed elevations of Florentine buildings. Here too his social status and personal connections were helpful, and he was able to organize personal seminars with the mathematician Paolo dal Pozzo Toscanelli to even better understand the principles of geometry and perspective. The Florentine government began granting him small commissions to review structural components on the ongoing construction of the Cathedral, the first step in his gradual abandonment of sculpting altogether. He first accepted somewhat less prestigious commissions to redesign fortifications in towns near Florence, although more grandiose commissions would quickly follow: from the Hospital of the Innocents (an orphanage for illegitimate children) to culminate with the commission to plan the dome of the Cathedral (beating out his rival Ghiberti in the tender). From then on he would become one of the most celebrated architects of his day, and his commissions ranged from the headquarters of knightly orders (the Palagio di Parte Guelfa) to chapels for private citizens (like the Sagrestia Vecchia) and private residences (like the one belonging to the Medici).
Hopefully that gives you an overview of Brunelleschi's early life and background before becoming the leading architect in Florence. You can always pick up a biography if if you'd like to read more; if you can read Italian I'd suggest the book where I learned pretty much everything I know about him: Brunelleschi by Elena Capretti (Giunti Editore, 2003). I honestly don't know what his definitive english biography is, but Brunelleschi's Dome: How a Renaissance Genius Reinvented Architecture By Ross King is a bestseller from 2000 you could look at, although I haven't read it and by the title it sounds very Great Man-y.