The Optical fibre can transmit/receive any ethernet frames or packets faithfully for 5kms. However, the cables laid in the southern cross i.e. under the sea beds must be several miles long. The question is how they can transmit the signals faithfully. There is a limitation with the Cables. Next, I believe they put some kind of repeater. If so, how can they install repeater under the sea?
This is the kind of question that books, lecture courses etc are made of. A stack exchange answer can only give a very brief overview.
The Optical fibre can transmit/receive any ethernet frames or packets faithfully for 5kms.
That statement on it's own is meaningless. The distance you can transmit depends on both the type of transciever, the type of fiber, and whether or not amplification is used.
Using top end "pluggable" transcievers on modern single mode fiber without amplification I understand the limit is about 120km.
The question is how they can transmit the signals faithfully.
A combination of complex modulation schemes, high transmit powers and high quality glass can get you a long way but it can't get you intercontinental distances.
Next, I believe they put some kind of repeater.
In the old days they did indeed do optical-electrical-optical regeneration under the sea but that has fallen out of favour.
Nowadays they amplify signals optically with EDFAs and/or Raman amplifiers. EDFAs provide a point amplification, while Raman amplifiers beam a laser backwards down the infrastructure fiber which interacts with and amplifies the incoming optical signals.
A massive advantage of amplifying optically is it allows amplifying of wavelength division multiplexed signals without needing to demultiplex them. This is what has allowed the massive capacities of modern undersea cables to be achieved.
Still the optical amplifiers need power. This is provided by high voltage lines inside the cable. The actual fibers are a tiny part of a transoceanic cable.
If so, how can they install repeater under the sea?
Afaict they build them into the cable before they install it.
P.S. You might like to watch https://www.youtube.com/watch?v=__wn9zXFiy8