This is a basic issue of technological advancement. The purpose of the big horn was amplification. The sound produced by the needle on the discs was amplified to an audible level by getting piped through the horn (if you want a sense of what it would be like without one, you can easily enough - just through a record on a record player without the speakers turned on and lean in real close. You'll faintly hear the music if you listen very carefully).

The horn stopped being used because it became obsolete as a technology. Speakers are almost as old as phonograph technology, but it was slower to develop - initially people were using an air compression system that I admit I don't fully understand, but it was a lot simpler to just tack on a big horn and amplify the records acoustically - it was a more affordable and practical alternative. This makes sense from a quality standpoint as well. Instead of the sound quality being limited by expensive speaker technology, its ceiling was set by the quality of recording equipment and the ability of the cylinders (and later discs) to reproduce it - both of which were being pursued more aggressively by the competing record pioneers than the challenging early speaker technologies.

Speakers became more viable with a better understanding of electricity and the implementation of a system based on magnets, which are a necessary component of speakers and sound pick-up. As they began to surpass the horn as an amplifier (horns are notoriously limited by a fairly low max volume and a certain amount of inherent sound distortion), they were naturally adopted to either be local in record players or as external accessories. This change happened essentially around the time you see speakers emerging as a commercial technology in other areas. One of the earliest magnet-based speaker systems was patented in the 1920s, matching the popularization of radio. We also see the telephone reaching new achievements in this period, particularly the introduction of intercontinental and transatlantic phone service - people were getting better at moving sound across distances, and the point was rapidly reached where the horn could no longer compete with speakers in loudness or clarity, and died its natural death.

I can't give you firm date on when, exactly the first commercially available speaker-operated record player hit the market or when the death of the horn officially occurred - I'm less a historian of the phonograph than a hobbyist audiophile, and most of my knowledge in this comes from reading up on sound gear, not history. But I can tell you that the demise of the horn coincides with the increasing availability, affordability, and quality of speakers, which began in the 1920s.

The early phonographs were not electrical appliances. The needle directly vibrated a diaphragm which moved a column of air, producing sound. Although it is technically incorrect to say the horn amplifies the sound, you basically get the idea. This website explains it in pretty good detail: http://www.victor-victrola.com/Basics%20of%20the%20Acoustic%20Phonograph.htm

Later victrolas moved the horn inside the case, which was basically an aesthetic measure. The disadvantage of these mechanical phonographs was they were not loud, and the fidelity was poor. The advantage was it was entirely portable, and you did not need batteries.

More modern record players electronically amplify the signal. The vibrations of the needle are translated into an electric signal, which is amplified, and then drives speakers. As a result, it can be much louder and have better fidelity, making the old mechanical horn obsolete.