The short answer is that we don't know, but bread has been around for a long time. Bread predates agriculture - it's probable that one reason why people grew wheat and barley was because they already knew how to use it make bread. The oldest bread ever found (in Jordan) was about 14,000 years old. Bread does not survive well archaeologically, so it is likely that bread was made long before this.
How long before this? Grindstones have been found (e.g., in Italy), from about 30,000 years ago, that were used for grinding starchy plants. It's possible (but not certain) that bread was made, not necessarily from grains, but from other starchy plants. Bread appears to be of at least similar antiquity in Australia (again, not grains, but from various other seeds and starchy plants). It is quite possible that the invention of bread predates the settlement of Australia, and is over 60,000 years old.
Before the farming of wheat and barley in Europe, bread would not have been a staple food in Europe. The spread of cereal farming brought bread to the everyday table. Flatbreads predated leavened loaves of bread. For fluffy loaves, we need three things: the right varieties of wheat, yeast, and ovens. If the main grains are oats or barley or rye, breads are often flatbreads, and if they are leavened loaves, they are dense. Flatbread can be cooked in many ways, such as cakes in ashes, roasted, dry-fried on griddles, etc., while loaves are best cooked with ovens.
As for yeast, yeast is all over the place. However, yeast likes sugars, and most of the stuff in bread is starches and proteins and fibre that yeast can't eat. If you want to make yeast-bread, it's good to start with a source of yeast. This can be some of the bread from the previous batch - a sourdough starter (not just yeasts, but also various bacteria, which contribute to the flavour). Another source is yeast from beer. While beer is also made from grains, some of the starches are converted to sugars, which feed the yeast. Methods for doing this conversion include malting the grain (letting it germinate, and the embryo produces enzymes with convert starch to sugars) and chewing the grain (letting enzymes in saliva convert starch to sugars). Brewing beer and making bread in the same house? Just skim off some yeast from your fermenting beer, and mix into your bread dough, and it will rise. If you have an oven, you can bake a loaf in the oven, and if not, you can make a leavened flatbread.
The oldest definite evidence of leavened loaves of bread we have found are from ancient Egypt, with some bread surviving in tombs. However, it's likely that similar loaves were made earlier, since beer and grain were around in the area for thousands of years before then.
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