The penny-farthing was the standard until the diamond-frame "safety bike" was "invented" and revolutionalized the world of bikes.
It seems to me strange to have to "invent" to put the same size of wheels on a bike, composed of two triangles (the idea of triangles is meny hundreds of years old in civil engineering).
Why did those awkward and dangerous bikes exist?
It only seems strange to you because you've had the benefit of never having to learn the lessons of the penny farthing and the technological achievements of the safety bicycle firsthand (aka 'standing on the shoulders of giants syndrome').
The penny farthing is only awkward and dangerous because you are comparing it against a technological leap forward. That technology you are taking for granted, and are not realizing just how difficult it was to develop those technology without having the advantage of already knowing about it. Two of the key pieces of technology in a bicycle you are taking for granted:
JamesBradbury's answer shows the dandy horse.
The safety bicycle had some initial problems of its own too, it began with a treadle bicycle, prior to the inclusion of the chain drive.
Additionally, the idea of "triangles" and their use as structural members isn't as old as you might think. Sure, people have known for a long time that triangles are 'strong', but our analytical and mathematical knowledge of them is only since the middle 17th century.
So the bottom line: people rode penny farthings because the modern bicycle wasn't invented yet.
I think the simple reason is that the drivechain hadn't been invented.
In 1818 the dandy-horse or draisine was invented. This was similar in shape to a modern bicycle, but without pedals. Riders would scoot along with their feet.
If you want to go a bit faster, especially on any kind of incline, then you need a better way of putting power in. Without a drivechain the only option is pedals connected directly to a wheel. The larger the wheel, the bigger the gearing. The circumference of these wheels was measured in inches and is why you may still hear gear ratios described in gear-inches.
Of course the large wheels were unwieldy and limited by the length of the rider's inside leg.
Don't forget, when the penny farthing was invented, horses were still a major mode of transport - when you see people on horseback regularly, the idea of sitting that high up doesn't seem unreasonable.
A recent view is that technology was less of a factor vs. sociological factors. The main sociological difference between the Penny-farthing and today's bicycle frame is the perspective of safety: the rider wants to feel safe or not.
In the times of Penny-farthing mastering a difficult to ride horse/vehicle was a major feat, and riders wanted to show of their skills. Riding a more difficult vehicle was a wish by riders.
Today's bicycle with the diamond frame mostly offers safety for the driver, by having a good view around the traffic, and a good chance to stop when needed (vs. efficiency, aerodynamics, speed, acceleration, etc., while recumbent and other designs do better in those aspects). Today riders want a bike that offers higher safety, vs. feeling unsafe.
The modern view of history of technology says about technology development that:
There's a lot of research done about bicycle development in History of Technology, mainly because researchers who pioneered these theories came from the Netherlands. The paper published by Bijker ("King of the road: the social construction of the safety bicycle.") is a major work that introduced the theory of social construction of technological systems.
You can find a short summary of Bijker's book at the bicycle research project. A Google search can give you a pdf of the paper. Note that Bijker's book is not considered "new" anymore, but rather "classic", and many new research goes beyond.
Speed. Without the complexity of gearing.
They were racing machines, pure and simple. There's one in the Glasgow transport museum with aerodynamic rim profile and flattened fork blades! I think it's about 20-21lbs, still quite respectable today.
The smooth ride of large wheels on rough roads was a plus, but the larger the wheel you could turn given your inside leg measurement, the faster you went.
In the UK, gearing is still expressed as the diameter of the equivalent ungeared wheel - you'll turn a 100 inch gear going downhill...
Gears would add weight, noise, complexity. Belts, friction, slip, lost energy. Chains ... OK, when the Reynolds chain was invented, it took over.
But chain wasn't that revolutionary. All Reynolds added was a gap between rows of links, which could ride on sprockets (different sized sprockets, for speed multiplication).
That basic style of chain was centuries old by then, and in a form that blows the mind if you think about the implications of making it... that silver thread in the first photo is a chain, probably 0.2mm thick, with links about 0.5 by 1mm, assembled by hand. And repaired by hand too. (Yes, that huge planetary object is a 1c piece)
As a watch spring ran down, the chain transferred its pull to a larger diameter on the fusee cone, to keep the pull and the watches rate constant. If you think our engineering skills are that much better today, read it and weep.
