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I hear that miners solve math problems to secure Bitcoins... what kinds of problems? And why are the solutions worth "money"?

Also, what does an exchange get in return for real money when they get Bitcoins? I am talking tangible assets?

I understand that fiat money is not backed by anything tangible, so obviously the exchange is a promise of stated value for a Bitcoin's vaule. I just don't understand what gives the Bitcoin it's value?

I suppose that with a national currency I understand that there is an agreement to value all things tangible by the labor rate it takes to produce it. Thereby tying value from one item to another making the currency valid. Obviously Bitcoin works the same way. So what is the labor that is making the currency value standardized to Bitcoin it's self.

I hope those questions make sense. Please let me know. Thanks.

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Bitcoins have a value simply because of supply and demand. Mathematical laws created a fixed supply of Bitcoins, and people's expectation that they can use them as a means of exchange or store of value created demand.

Exchanges simply bring buyers and sellers together. Buyers wish to purchase Bitcoins, either as a means of exchange or as a store of value. Sellers are willing to sell Bitcoins because they'd rather have fiat currency, either because that's what they need to pay bills or because they expect Bitcoins to drop in value. At some price, buyers and sellers will meet.

Miners secure the Bitcoin block chain by providing the proof of work that Bitcoin uses to verify Bitcoin transactions. They receive a Bitcoin reward for doing this because that's how Bitcoin was designed.

Bitcoins have value because they are scarce and can easily by transferred over long distances. They also have value because there's a reasonable expectation that they'll be usable as a means of exchange and as a store of value. There really is no other requirement for a currency. Anything scarce, durable, divisible and fungible can be used as a currency if people agree to use it that way, and many people have agreed to use Bitcoins that way.

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Neither fiat currencies nor Bitcoin is backed by anything tangible - you cannot through any mechanism expect to be able to exchange them for a fixed quantity of any other good or combination of goods.

Neither is there any agreement associated with fiat or Bitcoin to value anything by the amount of labour it took to produce it.

Bitcoin and fiat currencies have a lot in common when it comes to what gives them value; properties that make them suitable as medium exchange, (possibly) store of value and (possibly) unit of accounts: durability, low transaction costs, irreversibility (for fiat only some transactions are irreversible, mainly cash), security, and scarcity.

The 'math problems' you refer to are hash computations. They don't have value outside of the Bitcoin network, but they add value to the currency because they make the Bitcoin system secure against double-spending (someone trying to spend the same money more than once). Just like when you accept fiat currency notes you do not get any value specifically from the watermarks, but the application of watermarks add security to the currency - and therefore makes it more valuable to people.

Neither Bitcoin nor fiat currencies would have any value, however, were it not for a coordination of people who agree to use them as money. For fiat currencies this coordination happens through a government monopoly on issuing such money, and insistence that taxes are paid using the monopoly fiat currency.

For bitcoin this coordination has happened because it was a first mover in the technological innovation that it represents: bringing about lower transaction fees (irrespective of geographical distance), faster payments, irreversibility, pseudonomity, and very importantly a lack of reliance on any central party (it is distributed).

Bitcoin will only be able to sustain this coordination (and its value) if there are significant network effects, which makes it difficult for new entrants in the cryptocurrency market to gain market share, and there is no technological innovation which makes Bitcoin inferior to a new payment mechanism.

The initial and current value of Bitcoin, starting from zero as it did, is based on people recognising the above features and also that the scarcity of the coin could make it appreciate in value because supply is limited and at some stage effectivley stops growing.

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David's answer explains nicely why people consider Bitcoins valuable, but I don't think it addresses all your questions directly.

Bitcoin's value doesn't originate from people willing to pay for the hashes performed by mining. That would be like saying that gold has value because people value drilling in the ground.

It has value because of its utility, not because of its origin. The mining payout scheme is just how Bitcoins are (initially) distributed.

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