Salt did not used to be worth its weight in gold. In fact, salt consumption used to be a lot higher than it is now due to the need to preserve whether by packing it in salt or pickling. This need drove salt to be a valuable commodity, and it was often under some sort of central control as a method of taxation on the population.

One example is the Gabelle system in France, which was both a salt tax and a mandatory duty for some citizens to buy salt, depending on the province. For example, people ages 8 and up under the Pays de grandes gabelles, which would be people living in Île-de-France, Berry, Orléanais, Touraine, Anjou, Maine, Bourbonnais, Normandy, Bourgogne, Champagne, and Picardy (so, a significant chunk of France) were obligated to purchase about 7 kg of salt per year from authorized sources, which was taxes heavily and used as a major source of revenue. People in this specific group were actually not allowed to use this salt to make salted products. This system existed in some form or another from the 1300s to 1946. Salt prices varied wildly from region to region, and it was about 60 times more expensive in some areas than others as you can see here. Someone else can take on the challenge of explaining the relative value of a livre, but on the next page you can see the average consumption of salt per person being between 10 and 20 pounds.

The salt monopoly in China was just ended in 2014 after some 2600 years in service, and at times it made up over 80% of revenue for their government (note: this would be in a time where a direct "income tax" or the like was not feasible - most taxation was done in terms of conscripting labor from the peasantry). You can read about the price and average consumption of salt in 1901 in China here, which suggests that salt purchases were a real material hardship on some people in some areas, but not most people, with salt being more than 20x more expensive in some areas than others. In fact, in many areas a person could buy a year's worth of salt for a half-day's work (see page 40-45), so, like today, for many people salt was not worth much, so to speak. This paper details the sources and distribution methods of salt and explains why it was so much more expensive in some areas. Average salt consumption was about 5 kg per year.

According to the CDC, the average American eats about 1.25 kg of salt a year, so you might wonder why people in the past were using so much salt. A very significant amount of salt was being used to preserve food (recipes I've seen call for more than 1 kg of salt to preserve 12 kg of pork), as well as for agricultural purposes (feeding salt to animals). After canning and especially refrigeration took off, salt consumption fell, because people simply didn't need it.

If you're curious about this topic, I highly recommend the book Salt by Mark Kurlansky.

As to where salt comes from, salt today pretty much comes from where it has always came from, which is either mining or dried brine. Salt is so much cheaper today because, for one, technology to mine and dry seawater has become more efficient over time, and also the average salt consumption has dropped, since with the advent of refrigeration and canning salt is no longer needed to preserve food.

The whole concept of salt being 'worth its weight in gold' is a gross exaggeration. Not sure about the ancient times (maybe someone else would cover that period) but at least since the High Middle Ages salt was never even close to being exceedingly expensive commodity. Let me invoke two examples:

In 1473, a burgher from Warsaw (Poland) bought 100 barrels of salt for 100 marks. This gives us one mark per barrel. One barrel was equal to four hundredweight, i.e. 640 pounds or 290 kg (given the place of a trade I am assuming that Warsaw hundredweight, equal to 160 pounds was used). One mark was equal to half a pound (220 g) of silver and was divided to 48 silver groshes (basic monetary unit for everyday use, similar to English shilling, abbreviated 'gr' below), equal to 4.5 grams of silver [g(Ag) for short] or 1/6th ounce of silver).

So, it seems that for 4.5 grams of silver (1 gr) one was able to buy 6 pounds of salt [lb(NaCl) for short]. This means that 1 lb of salt had a cost of 1.25 g(Ag). Sure, we're speaking about the bulk trade here, but in the first decade of XVI century, salt has been sold at the salt mines of Krakow at a price of 8 gr per hundredweight (0.23 gAg per 1 lb of salt).

The price above means that merchants were selling salt at approximately 300% markup, what is understandable, as on a 200-250 miles trip from the mine to the city, there were at least several toll stations, and the tolls for salt tended to be very high.

Now, let's move back a century. In 1377 Wroclaw (since 1335 within the borders of Bohemia), the price of one measure (15.07 litre) of high quality salt from Krakow mines was 16 gr. Given the average density of coarse halite is 2.7 g/cm^3, after slight rounding down we get 72 g(Ag) per 72 lbs[NaCl], what gives us again 1 g(Ag) per 1 lb(NaCl).

Was it a lot? Not really. Weekly income of a master craftsman in the 1370's Wroclaw was roughly 6 gr, one of a river toll collector - 3 gr. A pig could have been bought for 8 gr, gallon of beer - 2.5 gr, an axe - 5 gr. So, while the price of salt was obviously much higher than today, but was comparable with beer or other comestibles. It is interesting that it hasn't changed much between the late 14th century and early 16th century.

As u/Pabst_Blue_Gibbon wrote, the demand for salt decreased significantly with the advent of canning and plummeted when refrigeration became the norm. Additionally, the technological advancement made it possible to mine, evaporate and transport salt much more efficiently. Additionally, colonial expansion led to discovery of large salt flats and salterns, further increasing supply.

I have met the misconception that the salt must have been very valuable, because it has been used as a currency in ancient times. It isn't necessarily so, because barter was pretty normal procedure then, and salt, being impervious to any kind of spoiling, was very suitable to be a long-time investment.

And in case one wonders how the price of salt corresponded to the price of gold, a made a necessary calculation for the period and area I wrote about above (13th-14th century Poland). Most common gold coin was ducat, containing 3,5 g of gold and equal to 32 gr or 144 g(Ag). So, 1 g(Au) was equal to 41 g(Ag) and thus to approx. 40 lbs of salt. It means that weight-wise, gold was 18160 times more valuable than salt.