There isn't really one single inventor we can point to. Putting words with images in various ways is apparently a very natural process, as it has developed separately in multiple civilisations.

We can very roughly trace the history of the speech balloon in modern comics through the European use of speech scrolls, so called because artists would put paper scrolls in paintings and paint words on them. You've likely seen these before, since they were very common in religious and historical iconography until roughly the renaissance, when realism started to become emphasised. Even after that, they remained popular in other use for another century or two.

Eventually, they started appearing in political imagery. Both of the two most important cartoonists in that era, to whom we owe a lot in modern cartoons, occasionally used speech bubbles: William Hogarth, who pioneered early political cartoons and sequential narrative in modern art, was using them as early as the 1720s; and James Gillray, who established the political cartoon as a genre later in the 18th century employed them on and off.

Meanwhile, the speech scroll evolved into the banderole on both sides of the Atlantic, also frequently for political purposes. They were very common in propaganda pieces from the time, and by the time that Gillray used them, they already worked much in the same way that they do today. There's a very famous example from the Napoleonic Wars, when Jefferson imposed an embargo on virtually all non-US ships. You can find that at the bottom of this page, which also has a great example directly above it on how speech bubbles were used in the same manner as nowadays: https://columbia1893.weebly.com/jefferson-political-cartoons.html (Note: I know nothing of the page itself, just thought the collection of images nicely displays the topic.)

Finally, we can arrive at modern comics. In this case, it was mostly popularised by early modern American comics, which featured them extensively. The Katzenjammer Kids (Dirks) and The Yellow Kid (Outcault) are usually considered the first modern newspaper comic strips, and they both featured extensive use of speech bubbles, which the Americans then called "word balloons". The Little Bears (Swinnerton), Happy Hooligan (Opper), and Little Nemo (McCay) are other examples of popular strips in the 1890s and 1900s that made frequent use of speech bubbles. Almost all famous American comics after that used them - with a few notable exceptions such as Val Foster's Prince Valiant.

Europe was slower to catch on, with some exceptions. Alain Saint-Ogan, who makes a strong claim to the title of the pioneer of modern French comics, did use them a lot, as did Hergé - arguably the most famous comic artist in Europe ever. Both used them as early as in the 1920s, but they didn't become the norm until after World War II, when American comics started to become popular in Europe.