Broadly speaking, the modern drum kit set-up comes out of big band swing music in the 1930s. The idea of one person playing more than one drum at once was still relatively new at this point - drums had usually been associated with marching bands where people individually carried and played one drum. However, with the rise of silent film in the early 20th century, which was often accompanied by musicians, percussionists became much in demand in order to replicate sound effects. The percussionists doing this began to formulate a kit of drums which sat on the floor, which would allow them to replicate particular types of sounds (involving the use of the bass drum via a mallet connected to a pedal, to which other drums were connected in various ways). The percussion that ended up being used was a synthesis of many different traditions, such as the bass drum and snare from the Western marching band tradition, Turkish cymbals (the Zildjian company was founded in 1623, back in the relatively early days of the Ottoman Empire), and Eastern tom-toms.

In quite early pictures of jazz swing drummers playing drum sets in the 1920s and 1930s, you can see tom-toms that obviously look like Eastern-style drums, because they have not been yet reshaped to be cylindrical in shape like the bass drum and snare. So if you see the drummer in this picture of Bix Biederbicke's Wolverines from 1924 here, you can see an Eastern-style tom, and...well, the cymbal hangs from a string? At about 4 minutes into this 1932 performance of Cab Calloway's 'Hi De Ho' (which Blues Brothers fans might be familiar with), you can see a clear shot of Cozy Cole's drum set with the Eastern-style toms, and it does not yet look exactly like a modern drum set (though it does appear to have the bass drum with pedal, and a snare drum in the right place - and maybe also a hi-hat?).

Gene Krupa is usually seen as being particularly influential in making people see the potential of the drum kit as a rhythmic device within a band format. He first recorded in 1927, with McKenzie and Condon's Chicagoans (here's their 'Nobody's Sweetheart'), and generally became the model for the playing style for white swing bands (along with Chick Webb, who was more influential among black swing bands, but who I don't think there is surviving footage of, but who passed away in 1939, and who became established playing earlier style kits). By 1934 Krupa was playing with Benny Goodman's orchestra, one of the more successful and feted of the age, and Krupa's extroverted drumming was seen as one of the driving forces behind the success of Goodman's orchestra amongst young white people, who appreciated the rhythmic intensity.

Krupa is also strongly associated with the Slingerland Radio King drum kit, introduced in 1935. Along with the Gretsch Broadkaster, also introduced the same year, it is amongst the very first modern drum kits in the sense you mean...almost (neither advertise a hi-hat as being part of the kit). Previous Slingerland drum kits before 1935 were still old-style (for example, this picture from a catalogue here) but Slingerland used Krupa in its ads for the Radio King, and it seems to have quickly become the standard drum kit in the era. A Ludwig catalogue from 1935 has a professional drum kit with a bass drum, a snare, two cymbals, and an Eastern-style tom - but no hi-hat or set of modern toms. This same basic setup is advertised in a 1937 catalogue, but the front page of the 1939 catalogue has a picture of Paul Whiteman orchestra drummer Ray Bauduc playing Ludwig's first modern drum kit in what is transparently "we have one now!".

You can see Krupa play a Radio King in this 1937 footage of the Benny Goodman band playing 'Sing Sing Sing!' for a Hollywood movie, with the arrangement spotlighting his drumming in several places, and where he appears to be also employing a hi-hat to keep time. You can also see a young Buddy Rich using a Slingerland drum kit in 1939 playing 'Traffic Jam' with Artie Shaw and you can see Moe Purtill using a similar kit with the Glenn Miller orchestra in 1941, both prominently using hi-hats.

With the addition of the hi-hat to the Slingerland Radio King/the Gretsch Broadkaster by the likes of Krupa, the classic drum kit was born.