What was the value of the spice trade during the age of exploration?

It was extremely valuable. Cinnamon, for example, was worth a king's ransom. Literally. Cleopatra had a large stack of it. When she suggested she would burn it, it got Augustus' immediate attention. In Holland we have the expression 'peperduur' = 'as expensive as pepper. Spices were definitely more expensive than gold in those days.

Why were they so expensive?

Exclusivity: Spices - those spices - don't grow in Europe. They are lightweight, very long lasting and easy to transport. In other words: a caravan loaded with spices was economically worth the travel from (say) India or even China all the way to Europe.

Logistics: It's not as simple as loading a camel with spices and a handler walking next to it. You wouldn't want to carry a few blocks of gold on your back and walk through a bad neighborhood. You need a substantial guard to protect it. A lot of infantry and cavalry surrounding the caravan. All those animals and personnel had to be fed and accommodated, along the route. This adds a lot to the costs.

As you can see, it was the cost of getting it from where it was grown to where it would be consumed that made spices so valuable.

Okay, so spices were really expensive. Why the demand for it? Couldn't they use something else?

Of course. Most people couldn't afford such luxury anyway. Which is the reason why original Dutch food is considered very bland. We Dutch sold those spices! You don't eat your own merchandise, do you?

In those days preservation of food was difficult. I'm not a cook, but smoking and salting were the most popular and common methods. A lot of food we consider today as spoiled and throw away was eaten back then. Of course people didn't like the taste of spoiled meat. Adding spices to it disguised the taste of spoiled meat and made it actually taste good. Even in minute quantities.

This is an image of an antique pepper spoon. Species were that expensive.

Why did the prices eventually drop?

Production was at first intentionally limited. The Dutch VOC had a monopoly, and enforced it strictly. Later more companies (British East India Company, others) broke that monopoly by doing the same thing for their markets. Which lowered prices. British merchants didn't have to buy expensive Dutch pepper anymore. But the EIC couldn't keep the prices at the same level as the VOC did, otherwise merchants wouldn't be buying theirs. They had to be somewhat competitive.

Much later more was grown and better transportation (larger sailing ships and steamships) made it easier available. Higher speed was not important, but bigger carrying capacity was.

Try this link: The Luxury Trades of the Silk Road: How Much Did Silks and Spices Really Cost?

Here is my main source for the following answers. EDIT 6/4/2015: I have expanded this answer to elaborate on a number of things.

How much, in today terms, were they worth?

Around the year 1500, a quintal of pepper in Lisbon was worth up to 38 ducats. A ducat was 3.5g of gold and a quintal was only 60 grams of pepper... So, pepper was worth a bit more than twice its weight in gold! Other spices like cloves, cinnamon or nutmeg were probably worth significantly more than that.

And what is the origin of such extraordinary demand?

Spices were popular then for the same reasons they are today. But obviously they were more of a luxury item at that time. They weren't expensive so much because of a huge demand, but more fundamentally because of the problems of supply. It was very limited, and very far away, in a period where transportation was expensive.

A big part of this expense was related not just the technical limits of sailing ships and navigation, but also to protection costs (security against piracy and the like). And before the Industrial Revolution there was not much demand for European exports in the East. That made all Asian imports, including spices, expensive.

When and why did the price eventually drop?

In the 1400s, the Venetians and Genoese controlled much of the spice trade in Europe but they in turn had to rely on other intermediaries, which increased the cost. The Portuguese started to establish trading posts in Asia around 1500. Then came the Dutch and the British in the 1600s. The increasing access of European traders to the sources of the spices was a key factor that helped to bring down the price from that point onwards, increasing competition and eventually reducing protection costs.

Meanwhile a number of things increased the purchasing power of European traders. That included 1) huge imports of silver from the Americas in the 1500s 2) the Industrial Revolution, which allowed Europe to produce cheaper textiles and other goods and 3) colonization of India and other places, which gave British and other traders access to new trade items, most notably opium. All of those things would have helped to make spices and other Asian imports much cheaper in relative terms.