What gives bitcoins value? A simple explanation please 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.
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.